product integral
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A product integral is any
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
-based counterpart of the usual sum-based
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
. The first product integral ('' Type I'' below) was developed by the mathematician
Vito Volterra Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis. Biography Born in An ...
in 1887 to solve systems of
linear differential equation In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...
s. A. Slavík
''Product integration, its history and applications''
, Matfyzpress, Prague, 2007.
Other examples of product integrals are the geometric integral ('' Type II'' below), the bigeometric integral ('' Type III'' below), and some other integrals of non-Newtonian calculus. Michael Grossman
''The First Nonlinear System of Differential And Integral Calculus''
, 1979.
Michael Grossman
''Bigeometric Calculus: A System with a Scale-Free Derivative''
, 1983.
Product integrals have found use in areas from
epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and determinants of health and disease conditions in a defined population. It is a cornerstone of public health, and shapes policy decisions and evi ...
(the
Kaplan–Meier estimator The Kaplan–Meier estimator, also known as the product limit estimator, is a non-parametric statistic used to estimate the survival function from lifetime data. In medical research, it is often used to measure the fraction of patients living ...
) to stochastic
population dynamics Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems. History Population dynamics has traditionally been the dominant branch of mathematical biology, which has a ...
using multiplication integrals (multigrals),
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
and
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. The geometric integral, together with the geometric derivative, is useful in
image analysis Image analysis or imagery analysis is the extraction of meaningful information from images; mainly from digital images by means of digital image processing techniques. Image analysis tasks can be as simple as reading bar coded tags or as soph ...
Luc Florack and Hans van Asse
"Multiplicative calculus in biomedical image analysis"
Journal of Mathematical Imaging and Vision, , 2011.
and in the study of growth/decay phenomena (e.g., in
economic growth Economic growth can be defined as the increase or improvement in the inflation-adjusted market value of the goods and services produced by an economy in a financial year. Statisticians conventionally measure such growth as the percent rate o ...
,
bacterial growth 250px, Growth is shown as ''L'' = log(numbers) where numbers is the number of colony forming units per ml, versus ''T'' (time.) Bacterial growth is proliferation of bacterium into two daughter cells, in a process called binary fission. Providing ...
, and
radioactive decay Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is consid ...
). Agamirza E. Bashirov, Emine Misirli, Yucel Tandogdu, and Ali Ozyapic
"On modelling with multiplicative differential equations"
Applied Mathematics – A Journal of Chinese Universities, Volume 26, Number 4, pages 425–428, , Springer, 2011.
The bigeometric integral, together with the bigeometric derivative, is useful in some applications of
fractals In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...
,Marek Rybaczuk, Alicja Kedzia and Witold Zielinski (2001
"The concept of physical and fractal dimension II. The differential calculus in dimensional spaces"
''Chaos, Solitons, & Fractals''Volume 12, Issue 13, October 2001, pages 2537–2552.
and in the theory of elasticity in economics. This article adopts the "product" \prod notation for product integration instead of the "integral" \int (usually modified by a superimposed "times" symbol or letter P) favoured by
Volterra Volterra (; Latin: ''Volaterrae'') is a walled mountaintop town in the Tuscany region of Italy. Its history dates from before the 8th century BC and it has substantial structures from the Etruscan, Roman, and Medieval periods. History Volt ...
and others. An arbitrary classification of types is also adopted to impose some order in the field.


Basic definitions

The classical
Riemann integral In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of GÃ ...
of a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
f: ,bto\mathbb can be defined by the relation :\int_a^b f(x)\,dx = \lim_\sum f(x_i)\,\Delta x, where the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
is taken over all
partitions Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of ...
of the interval ,b/math> whose norms approach zero. Roughly speaking, product integrals are similar, but take the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of a
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
instead of the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of a sum. They can be thought of as "
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
" versions of " discrete"
products Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
. The most popular product integrals are the following:


Type I: Volterra integral

:\prod_a^b \big(1 + f(x)\,dx\big) = \lim_ \prod \big(1 + f(x_i)\,\Delta x\big). The type I product integral corresponds to
Volterra Volterra (; Latin: ''Volaterrae'') is a walled mountaintop town in the Tuscany region of Italy. Its history dates from before the 8th century BC and it has substantial structures from the Etruscan, Roman, and Medieval periods. History Volt ...
's original definition. The following relationship exists for
scalar function In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity ( ...
s f: ,b\to \mathbb: :\prod_a^b \big(1 + f(x)\,dx\big) = \exp\left(\int_a^b f(x) \,dx\right), which is not a multiplicative operator. (So the concepts of product integral and multiplicative integral are not the same). The Volterra product integral is most useful when applied to matrix-valued functions or functions with values in a
Banach algebra In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
, where the last equality is no longer true (see the references below). When applied to scalars belonging to a non-commutative field, to matrixes, and to operators, i.e. to mathematical objects that don't commute, the Volterra integral splits in two definitions Left Product integral :P(A,D)=\prod_^(\mathbb+A(\xi_i)\Delta t_i) = (\mathbb+A(\xi_m)\Delta t_m) \cdots (\mathbb+A(\xi_1)\Delta t_1) With the notation of left products (i.e. normal products applied from left) :\prod_a^b (\mathbb+A(t)dt)=\lim_ P(A,D) Right Product Integral :P(A,D)^*=\prod_^(\mathbb+A(\xi_i)\Delta t_i) = (\mathbb+A(\xi_1)\Delta t_1) \cdots (\mathbb+A(\xi_m)\Delta t_m) With the notation of right products (i.e. applied from right) :(\mathbb+A(t)dt) \prod_a^b =\lim_ P(A,D)^* Where \mathbb is the identity matrix and D is a partition of the interval ,bin the Riemann sense, i.e. the limit is over the maximum interval in the partition. Note how in this case time ordering comes evident in the definitions. For scalar functions, the derivative in the Volterra system is the
logarithmic derivative In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function ''f'' is defined by the formula \frac where f' is the derivative of ''f''. Intuitively, this is the infinitesimal relative change in ''f'' ...
, and so the Volterra system is not a multiplicative calculus and is not a non-Newtonian calculus.


Type II: geometric integral

:\prod_a^b f(x)^ = \lim_ \prod = \exp\left(\int_a^b \ln f(x) \,dx\right), which is called the geometric integral and is a multiplicative operator. This definition of the product integral is the
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
analog of the discrete
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
operator :\prod_^b (with i, a, b \in \mathbb) and the multiplicative analog to the (normal/standard/ additive)
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
:\int_a^b dx (with x \in ,b/math>): : It is very useful in
stochastics Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselve ...
, where the
log-likelihood The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood functi ...
(i.e. the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
of a product integral of
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s) equals the
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
of these (
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
ly many)
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s: :\ln \prod_a^b p(x)^ = \int_a^b \ln p(x) \,dx.


Type III: bigeometric integral

:\prod_a^b f(x)^ = \exp\left(\int_r^s \ln f(e^x) \,dx\right), where ''r'' = ln ''a'', and ''s'' = ln ''b''. The type III product integral is called the bigeometric integral and is a multiplicative operator.


Results

;Basic results The following results are for the type II product integral (the geometric integral). Other types produce other results. : \prod_a^b c^ = c^, : \prod_a^b x^ = \frac ^, : \prod_0^b x^ = b^b ^, : \prod_a^b \left(f(x)^k\right)^ = \left(\prod_a^b f(x)^\right)^k, : \prod_a^b \left(c^\right)^ = c^, The geometric integral (type II above) plays a central role in the geometric calculus, M. Grossman, R. Katz
''Non-Newtonian Calculus''
, Lee Press, 1972.
which is a multiplicative calculus. The inverse of the geometric integral, which is the geometric derivative, denoted f^*(x), is defined using the following relationship: : f^*(x)=\exp\left(\frac\right) Thus, the following can be concluded: ; The fundamental theorem : \prod_a^b f^*(x)^ = \prod_a^b \exp\left(\frac \,dx\right) = \frac, ;
Product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
: (fg)^* = f^* g^*. ;
Quotient rule In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h(x)=f(x)/g(x), where both and are differentiable and g(x)\neq 0. The quotient rule states that the deriva ...
: (f/g)^* = f^*/g^*. ;
Law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...
: \sqrt \underset \prod_x X^, where X is a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
with
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
''F''(''x''). Compare with the standard
law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...
: : \frac \underset \int X \,dF(x).


Lebesgue-type product-integrals

Just like the Lebesgue version of (classical) integrals, one can compute product integrals by approximating them with the product integrals of simple functions. Each type of product integral has a different form for simple functions.


Type I: Volterra integral

Because simple functions generalize
step function In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having onl ...
s, in what follows we will only consider the special case of simple functions that are step functions. This will also make it easier to compare the Lebesgue definition with the Riemann definition. Given a
step function In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having onl ...
f: ,b\to \mathbb with corresponding partition a = y_0 < y_1 < \dots < y_m and a tagged partition : a = x_0 < x_1 < \dots < x_n = b, \quad x_0 \le t_0 \le x_1, x_1 \le t_1 \le x_2, \dots, x_ \le t_ \le x_n, one
approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
of the "Riemann definition" of the type I product integral is given byA. Slavík
''Product integration, its history and applications''
p. 65. Matfyzpress, Prague, 2007. .
: \prod_^ \left \big(1 + f(t_k)\big) \cdot (x_ - x_k) \right The (type I) product integral was defined to be, roughly speaking, the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of these
products Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
by
Ludwig Schlesinger Ludwig Schlesinger (Hungarian: Lajos Schlesinger, Slovak Ľudovít Schlesinger), (1 November 1864 – 15 December 1933) was a German mathematician known for the research in the field of linear differential equations. Biography Schlesinger att ...
in a 1931 article. Another approximation of the "Riemann definition" of the type I product integral is defined as : \prod_^ \exp\big(f(t_k) \cdot (x_ - x_k)\big). When f is a
constant function In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image). Basic propertie ...
, the limit of the first type of approximation is equal to the second type of approximation.A. Slavík
''Product integration, its history and applications''
p. 71. Matfyzpress, Prague, 2007. .
Notice that in general, for a step function, the value of the second type of approximation doesn't depend on the partition, as long as the partition is a refinement of the partition defining the step function, whereas the value of the first type of approximation ''does'' depend on the fineness of the partition, even when it is a refinement of the partition defining the step function. It turns out thatA. Slavík
''Product integration, its history and applications''
p. 72. Matfyzpress, Prague, 2007. .
that for ''any'' product-integrable function f, the limit of the first type of approximation equals the limit of the second type of approximation. Since, for step functions, the value of the second type of approximation doesn't depend on the fineness of the partition for partitions "fine enough", it makes sense to defineA. Slavík
''Product integration, its history and applications''
p. 80. Matfyzpress, Prague, 2007.
the "Lebesgue (type I) product integral" of a step function as : \prod_a^b \big(1 + f(x) \,dx\big) \overset \prod_^ \exp\big(f(s_k) \cdot (y_ - y_k)\big), where y_0 < a = s_0 < y_1 < \dots < y_ < s_ < y_n = b is a tagged partition, and again a = y_0 < y_1 < \dots < y_m is the partition corresponding to the step function f. (In contrast, the corresponding quantity would not be unambiguously defined using the first type of approximation.) This generalizes to
arbitrary Arbitrariness is the quality of being "determined by chance, whim, or impulse, and not by necessity, reason, or principle". It is also used to refer to a choice made without any specific criterion or restraint. Arbitrary decisions are not necess ...
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
s readily. If X is a measure space with measure \mu, then for any product-integrable simple function f(x) = \sum_^n a_k I_(x) (i.e. a
conical combination Given a finite number of vectors x_1, x_2, \dots, x_n in a real vector space, a conical combination, conical sum, or weighted sum''Convex Analysis and Minimization Algorithms'' by Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, 1993, pp. 101, 102 ...
of the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
s for some disjoint measurable sets A_0, A_1, \dots, A_ \subseteq X), its type I product integral is defined to be : \prod_X \big(1 + f(x) \,d\mu(x)\big) \overset \prod_^ \exp\big(a_k \mu(A_k)\big), since a_k is the value of f at any point of A_k. In the special case where X = \mathbb, \mu is
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wi ...
, and all of the measurable sets A_k are intervals, one can verify that this is equal to the definition given above for that special case. Analogous to the theory of Lebesgue (classical) integrals, the Volterra product integral of any product-integrable function f can be written as the limit of an increasing
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of Volterra product integrals of product-integrable simple functions. Taking
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
s of both sides of the above definition, one gets that for any product-integrable simple function f: : \ln \left(\prod_X \big(1 + f(x) \,d\mu(x)\big) \right) = \ln \left( \prod_^ \exp\big(a_k \mu(A_k)\big) \right) = \sum_^ a_k \mu(A_k) = \int_X f(x) \,d\mu(x) \iff : \prod_X \big(1 + f(x) \,d\mu(x)\big) = \exp \left( \int_X f(x) \,d\mu(x) \right), where we used the definition of integral for simple functions. Moreover, because
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s like \exp can be interchanged with limits, and the product integral of any product-integrable function f is equal to the limit of product integrals of simple functions, it follows that the relationship : \prod_X \big(1 + f(x) \,d\mu(x)\big) = \exp \left( \int_X f(x) \,d\mu(x) \right) holds generally for ''any'' product-integrable f. This clearly generalizes the property mentioned above. The Volterra product integral is multiplicative as a
set function In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R an ...
,Gill, Richard D., Soren Johansen
"A Survey of Product Integration with a View Toward Application in Survival Analysis"
The Annals of Statistics 18, no. 4 (December 1990): 1501—555, p. 1503.
which can be shown using the above property. More specifically, given a product-integrable function f one can define a set function _f by defining, for every measurable set B \subseteq X , : _f(B) \overset \prod_B \big(1 + f(x) \,d\mu(x)\big) \overset \prod_X \big(1 + (f \cdot I_B)(x) \,d\mu(x)\big), where I_B(x) denotes the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
of B. Then for any two ''disjoint'' measurable sets B_1, B_2 one has : \begin _f(B_1 \sqcup B_2) &= \prod_ \big(1 + f(x) \,d\mu(x)\big) \\ &= \exp\left( \int_ f(x) \,d\mu(x) \right) \\ &= \exp\left( \int_ f(x) \,d\mu(x) + \int_ f(x) \,d\mu(x) \right) \\ &= \exp\left( \int_ f(x) \,d\mu(x) \right) \exp\left( \int_ f(x) \,d\mu(x) \right) \\ &= \prod_ (1 + f(x)d \mu(x)) \prod_ (1 + f(x) \,d\mu(x)) \\ &= _f(B_1 ) _f(B_2). \end This property can be contrasted with measures, which are ''additive''
set function In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R an ...
s. However the Volterra product integral is ''not'' multiplicative as a
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...
. Given two product-integrable functions f , g, and a measurable set A, it is generally the case that : \prod_A \big(1 + (fg)(x) \,d\mu(x)\big) \neq \prod_A \big(1 + f(x) \,d\mu(x)\big) \prod_A \big(1 + g(x) \,d\mu(x)\big).


Type II: geometric integral

If X is a
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
with measure \mu, then for any product-integrable simple function f(x) = \sum_^n a_k I_(x) (i.e. a
conical combination Given a finite number of vectors x_1, x_2, \dots, x_n in a real vector space, a conical combination, conical sum, or weighted sum''Convex Analysis and Minimization Algorithms'' by Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, 1993, pp. 101, 102 ...
of the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
s for some disjoint measurable sets A_0, A_1, \dots, A_ \subseteq X), its type II product integral is defined to be : \prod_X f(x)^ \overset \prod_^ a_k^. This can be seen to generalize the definition given above. Taking
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
s of both sides, we see that for any product-integrable simple function f: : \ln \left( \prod_X f(x)^ \right) = \sum_^ \ln(a_k) \mu(A_k) = \int_X \ln f(x) \,d\mu (x) \iff \prod_X f(x)^ = \exp\left( \int_X \ln f(x) \,d\mu (x) \right), where we have used the definition of the Lebesgue integral for simple functions. This observation, analogous to the one already made above, allows one to entirely reduce the " Lebesgue theory of geometric integrals" to the Lebesgue theory of (classical) integrals. In other words, because
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s like \exp and \ln can be interchanged with limits, and the product integral of any product-integrable function f is equal to the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of some increasing
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of product integrals of simple functions, it follows that the relationship : \prod_X f(x)^ = \exp\left( \int_X \ln f(x) \,d\mu(x) \right) holds generally for ''any'' product-integrable f. This generalizes the property of geometric integrals mentioned above.


See also

*
List of derivatives and integrals in alternative calculi There are many alternatives to the Calculus, classical calculus of Isaac Newton, Newton and Gottfried Wilhelm Leibniz, Leibniz; for example, each of the infinitely many non-Newtonian calculi. Occasionally an alternative calculus is more suited th ...
*
Indefinite product In mathematics, the indefinite product operator is the inverse operator of Q(f(x)) = \frac. It is a discrete version of the geometric integral of geometric calculus, one of the non-Newtonian calculi. Some authors use term discrete multiplicative int ...
*
Logarithmic derivative In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function ''f'' is defined by the formula \frac where f' is the derivative of ''f''. Intuitively, this is the infinitesimal relative change in ''f'' ...
*
Ordered exponential The ordered exponential, also called the path-ordered exponential, is a mathematical operation defined in non-commutative algebras, equivalent to the exponential of the integral in the commutative algebras. In practice the ordered exponential is ...
*
Fractal derivative In applied mathematics and mathematical analysis, the fractal derivative or Hausdorff derivative is a non-Newtonian generalization of the derivative dealing with the measurement of fractals, defined in fractal geometry. Fractal derivatives were ...


References


External links


Non-Newtonian calculus website
* Richard Gill
''Product Integration''
* Richard Gill
''Product Integral Symbol''
* David Manura

* Tyler Neylon
''Easy bounds for n!''

An Introduction to Multigral (Product) and Dx-less Calculus


* Antonín Slavík
''An introduction to product integration''
* Antonín Slavík
''Henstock–Kurzweil and McShane product integration''
{{DEFAULTSORT:Product Integral Integrals Multiplication Non-Newtonian calculus