HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Pettis integral or Gelfand–Pettis integral, named after Israel M. Gelfand and Billy James Pettis, extends the definition of the
Lebesgue integral In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...
to vector-valued functions on 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 ...
, by exploiting duality. The integral was introduced by Gelfand for the case when the measure space is an interval with
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 higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
. The integral is also called the weak integral in contrast to the
Bochner integral In mathematics, the Bochner integral, named for Salomon Bochner, extends the definition of a multidimensional Lebesgue integral to functions that take values in a Banach space, as the limit of integrals of simple functions. The Bochner integral p ...
, which is the strong integral.


Definition

Let f : X \to V where (X,\Sigma,\mu) is a measure space and V is a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
(TVS) with a continuous dual space V' that separates points (that is, if x \in V is nonzero then there is some l \in V' such that l(x) \neq 0), for example, V is a
normed space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war p ...
or (more generally) is a Hausdorff
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
TVS. Evaluation of a functional may be written as a
duality pairing In mathematics, a dual system, dual pair or a duality over a Field (mathematics), field \mathbb is a triple (X, Y, b) consisting of two vector spaces, X and Y, over \mathbb and a non-Degenerate bilinear form, degenerate bilinear map b : X \times Y ...
: \langle \varphi, x \rangle = \varphi The map f : X \to V is called if for all \varphi \in V', the scalar-valued map \varphi \circ f is a measurable map. A weakly measurable map f : X \to V is said to be if there exists some e \in V such that for all \varphi \in V', the scalar-valued map \varphi \circ f is
Lebesgue integrable In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...
(that is, \varphi \circ f \in L^1\left( X, \Sigma, \mu \right)) and \varphi(e) = \int_X \varphi(f(x)) \, \mathrm \mu(x). The map f : X \to V is said to be if \varphi \circ f \in L^1\left( X, \Sigma, \mu \right) for all \varphi \in V^ and also for every A \in \Sigma there exists a vector e_A \in V such that \langle \varphi, e_A \rangle = \int_A \langle \varphi, f(x) \rangle \, \mathrm \mu(x) \quad \text \varphi \in V'. In this case, e_A is called the of f on A. Common notations for the Pettis integral e_A include \int_A f \, \mathrm\mu, \qquad \int_A f(x) \, \mathrm\mu(x), \quad \text~ A=X ~ \text \quad \mu To understand the motivation behind the definition of "weakly integrable", consider the special case where V is the underlying scalar field; that is, where V = \R or V = \Complex. In this case, every linear functional \varphi on V is of the form \varphi(y) = s y for some scalar s \in V (that is, \varphi is just scalar multiplication by a constant), the condition \varphi(e) = \int_A \varphi(f(x)) \, \mathrm \mu(x) \quad\text~ \varphi \in V', simplifies to s e = \int_A s f(x) \, \mathrm \mu(x) \quad\text~ s. In particular, in this special case, f is weakly integrable on X if and only if f is Lebesgue integrable.


Relation to Dunford integral

The map f : X \to V is said to be if \varphi \circ f \in L^1\left( X, \Sigma, \mu \right) for all \varphi \in V^ and also for every A \in \Sigma there exists a vector d_A \in V'', called the of f on A, such that \langle d_A, \varphi \rangle = \int_A \langle \varphi, f(x) \rangle \, \mathrm \mu(x) \quad \text \varphi \in V' where \langle d_A, \varphi \rangle = d_A(\varphi). Identify every vector x \in V with the map scalar-valued functional on V' defined by \varphi \in V' \mapsto \varphi(x). This assignment induces a map called the canonical evaluation map and through it, V is identified as a vector subspace of the double dual V''. The space V is a
semi-reflexive space In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) ''X'' such that the canonical evaluation map from ''X'' into its bidual (which is the strong dual of ''X'') is bijecti ...
if and only if this map is
surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
. The f : X \to V is Pettis integrable if and only if d_A \in V for every A \in \Sigma.


Properties

An immediate consequence of the definition is that Pettis integrals are compatible with continuous linear operators: If \Phi \colon V_1 \to V_2 is linear and continuous and f \colon X \to V_1 is Pettis integrable, then \Phi\circ f is Pettis integrable as well and \int_X \Phi(f(x))\,d\mu(x) = \Phi \left(\int_X f(x)\,d\mu(x) \right). The standard estimate \left , \int_X f(x)\,d\mu(x) \right , \leq \int_X , f(x), \, d\mu(x) for real- and complex-valued functions generalises to Pettis integrals in the following sense: For all continuous seminorms p\colon V\to\mathbb and all Pettis integrable f \colon X \to V, p \left (\int_X f(x)\,d\mu(x) \right ) \leq \underline p(f(x)) \,d\mu(x) holds. The right-hand side is the lower Lebesgue integral of a ,\infty/math>-valued function, that is, \underline g \,d\mu := \sup \left \. Taking a lower Lebesgue integral is necessary because the integrand p\circ f may not be measurable. This follows from the Hahn-Banach theorem because for every vector v\in V there must be a continuous functional \varphi\in V^* such that \varphi(v) = p(v) and for all w \in V, , \varphi(w), \leq p(w). Applying this to v := \int_X f \, d\mu gives the result.


Mean value theorem

An important property is that the Pettis integral with respect to a finite measure is contained in the closure of the
convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...
of the values scaled by the measure of the integration domain: \mu(A) < \infty \text \int_A f\,d\mu \in \mu(A) \cdot \overline This is a consequence of the Hahn-Banach theorem and generalizes the mean value theorem for integrals of real-valued functions: If V = \R, then closed convex sets are simply intervals and for f \colon X \to
, b The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math>, the following inequalities hold: \mu(A) a ~\leq~ \int_A f \, d\mu ~\leq~ \mu(A)b.


Existence

If V = \R^n is finite-dimensional then f is Pettis integrable if and only if each of f’s coordinates is Lebesgue integrable. If f is Pettis integrable and A\in\Sigma is a measurable subset of X, then by definition f_ \colon A\to V and f \cdot 1_A \colon X \to V are also Pettis integrable and \int_A f_ \,d\mu = \int_X f \cdot 1_A \,d\mu. If X is a topological space, \Sigma = \mathfrak_X its Borel-\sigma-algebra, \mu a
Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...
that assigns finite values to compact subsets, V is
quasi-complete In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete if every closed and bounded subset is complete. This concept is of considerable importance for non- metrizable TVSs. Properties * Eve ...
(that is, every ''bounded''
Cauchy net In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a function whose domain is a directed set. The codomain of this function is usually some topological space. Nets directly generalize ...
converges) and if f is continuous with compact support, then f is Pettis integrable. More generally: If f is weakly measurable and there exists a compact, convex C\subseteq V and a null set N\subseteq X such that f(X \setminus N) \subseteq C, then f is Pettis-integrable.


Law of large numbers for Pettis-integrable random variables

Let (\Omega, \mathcal F, \operatorname P) be a probability space, and let V be a topological vector space with a dual space that separates points. Let v_n : \Omega \to V be a sequence of Pettis-integrable random variables, and write \operatorname E _n/math> for the Pettis integral of v_n (over X). Note that \operatorname E _n/math> is a (non-random) vector in V, and is not a scalar value. Let \bar v_N := \frac \sum_^N v_n denote the sample average. By linearity, \bar v_N is Pettis integrable, and \operatorname E bar v_N= \frac \sum_^N \operatorname E _n\in V. Suppose that the partial sums \frac \sum_^N \operatorname E bar v_n/math> converge absolutely in the topology of V, in the sense that all rearrangements of the sum converge to a single vector \lambda \in V. The weak law of large numbers implies that \langle \varphi, \operatorname E bar v_N- \lambda \rangle \to 0 for every functional \varphi \in V^*. Consequently, \operatorname E bar v_N\to \lambda in the
weak topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
on X. Without further assumptions, it is possible that \operatorname E bar v_N/math> does not converge to \lambda. To get strong convergence, more assumptions are necessary.


See also

* * * * *


References

* James K. Brooks, ''Representations of weak and strong integrals in Banach spaces'',
Proceedings of the National Academy of Sciences of the United States of America ''Proceedings of the National Academy of Sciences of the United States of America'' (often abbreviated ''PNAS'' or ''PNAS USA'') is a peer-reviewed multidisciplinary scientific journal. It is the official journal of the National Academy of Scie ...
63, 1969, 266–270
Fulltext
* Israel M. Gel'fand, ''Sur un lemme de la théorie des espaces linéaires'', Commun. Inst. Sci. Math. et Mecan., Univ. Kharkoff et Soc. Math. Kharkoff, IV. Ser. 13, 1936, 35–40 *
Michel Talagrand Michel Pierre Talagrand (; born 15 February 1952) is a French mathematician working in probability theory, functional analysis and mathematical physics. Doctor of Science since 1977, he has been, since 1985, directeur de recherches at CNRS and a ...
, ''Pettis Integral and Measure Theory'', Memoirs of the AMS no. 307 (1984) * {{Functional analysis Functional analysis Integrals