In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Fatou's lemma establishes an
inequality relating the
Lebesgue integral
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
of the
limit inferior of a
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
functions to the limit inferior of integrals of these functions. The
lemma
Lemma may refer to:
Language and linguistics
* Lemma (morphology), the canonical, dictionary or citation form of a word
* Lemma (psycholinguistics), a mental abstraction of a word about to be uttered
Science and mathematics
* Lemma (botany), ...
is named after
Pierre Fatou
Pierre Joseph Louis Fatou (28 February 1878 – 9 August 1929) was a French mathematician and astronomer. He is known for major contributions to several branches of analysis. The Fatou lemma and the Fatou set are named after him.
Biography
...
.
Fatou's lemma can be used to prove the
Fatou–Lebesgue theorem
In mathematics, the Fatou–Lebesgue theorem establishes a chain of inequalities relating the integrals (in the sense of Lebesgue) of the limit inferior and the limit superior of a sequence of functions to the limit inferior and the limit sup ...
and Lebesgue's
dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary t ...
.
Standard statement
In what follows,
denotes the
-algebra of
Borel sets on
This_sequence_converges_uniformly_on_''S''_to_the_zero_function_and_the_limit,_0,_is_reached_in_a_finite_number_of_steps:_for_every_''x'' ≥ 0,_if_,_then_''f
n''(''x'') = 0.__However,_every_function_''f
n''_has_integral_−1.__Contrary_to_Fatou's_lemma,_this_value_is_strictly_less_than_the_integral_of_the_limit_(0).__
As_discussed_in__below,_the_problem_is_that_there_is_no_uniform_integrable_bound_on_the_sequence_from_below,_while_0_is_the_uniform_bound_from_above.
_Reverse_Fatou_lemma
Let_''f''
1,_''f''
2,_. . ._be_a_sequence_of_extended_real_number_line">extended_real-valued_measurable_functions_defined_on_a_measure_space_(''S'',''Σ'',''μ'')._If_there_exists_a_non-negative_integrable_function_''g''_on_''S''_such_that_''f''
''n'' ≤ ''g''_for_all_''n'',_then
:
\limsup_\int_S_f_n\,d\mu\leq\int_S\limsup_f_n\,d\mu.
Note:_Here_''g integrable''_means_that_''g''_is_measurable_and_that_
\textstyle\int_S_g\,d\mu<\infty.
_Sketch_of_proof
We_apply_linearity_of_Lebesgue_integral_and_Fatou's_lemma_to_the_sequence_
g_-_f_n.__Since_
\textstyle\int_Sg\,d\mu_<_+\infty,_this_sequence_is_defined_
\mu-almost_everywhere_and_non-negative.
_Extensions_and_variations_of_Fatou's_lemma
_Integrable_lower_bound
Let_''f''
1,_''f''
2,_. . ._be_a_sequence_of_extended_real-valued_measurable_functions_defined_on_a_measure_space_(''S'',''Σ'',''μ'')._If_there_exists_an_integrable_function_''g''_on_''S''_such_that_''f''
''n'' ≥ −''g''_for_all_''n'',_then
:
\int_S_\liminf__f_n\,d\mu
_\le_\liminf__\int_S_f_n\,d\mu.
_Proof
Apply_Fatou's_lemma_to_the_non-negative_sequence_given_by_''f''
''n'' + ''g''.
_Pointwise_convergence
If_in_the_previous_setting_the_sequence_''f''
1,_''f''
2,_. . ._Pointwise_convergence.html" ;"title="xtended_real_number_line.html" ;"title=",2n.html" ;"title=",∞) with the Borel σ-algebra and the Lebesgue measure. For every natural number ''n'' define
:
f_n(x)=\begin-\frac1n&\textx\in [n,2n">,∞) with the Borel σ-algebra and the Lebesgue measure. For every natural number ''n'' define
:
f_n(x)=\begin-\frac1n&\textx\in [n,2n\\
0&\text
\end
This sequence converges uniformly on ''S'' to the zero function and the limit, 0, is reached in a finite number of steps: for every ''x'' ≥ 0, if , then ''fn''(''x'') = 0. However, every function ''fn'' has integral −1. Contrary to Fatou's lemma, this value is strictly less than the integral of the limit (0).
As discussed in below, the problem is that there is no uniform integrable bound on the sequence from below, while 0 is the uniform bound from above.
Reverse Fatou lemma
Let ''f''1, ''f''2, . . . be a sequence of extended real number line">extended real-valued measurable functions defined on a measure space (''S'',''Σ'',''μ''). If there exists a non-negative integrable function ''g'' on ''S'' such that ''f''''n'' ≤ ''g'' for all ''n'', then
:
\limsup_\int_S f_n\,d\mu\leq\int_S\limsup_f_n\,d\mu.
Note: Here ''g integrable'' means that ''g'' is measurable and that \textstyle\int_S g\,d\mu<\infty.
Sketch of proof
We apply linearity of Lebesgue integral and Fatou's lemma to the sequence g - f_n. Since \textstyle\int_Sg\,d\mu < +\infty, this sequence is defined \mu-almost everywhere and non-negative.
Extensions and variations of Fatou's lemma
Integrable lower bound
Let ''f''1, ''f''2, . . . be a sequence of extended real-valued measurable functions defined on a measure space (''S'',''Σ'',''μ''). If there exists an integrable function ''g'' on ''S'' such that ''f''''n'' ≥ −''g'' for all ''n'', then
:
\int_S \liminf_ f_n\,d\mu
\le \liminf_ \int_S f_n\,d\mu.
Proof
Apply Fatou's lemma to the non-negative sequence given by ''f''''n'' + ''g''.
Pointwise convergence
If in the previous setting the sequence ''f''1, ''f''2, . . . Pointwise convergence">converges pointwise to a function ''f'' ''μ''-almost everywhere on ''S'', then
:\int_S f\,d\mu \le \liminf_ \int_S f_n\,d\mu\,.
Proof
Note that ''f'' has to agree with the limit inferior of the functions ''f''''n'' almost everywhere, and that the values of the integrand on a set of measure zero have no influence on the value of the integral.
Convergence in measure
The last assertion also holds, if the sequence ''f''1, ''f''2, . . . converges in measure to a function ''f''.
Proof
There exists a subsequence such that
:\lim_ \int_S f_\,d\mu=\liminf_ \int_S f_n\,d\mu.
Since this subsequence also converges in measure to ''f'', there exists a further subsequence, which converges pointwise to ''f'' almost everywhere, hence the previous variation of Fatou's lemma is applicable to this subsubsequence.
Fatou's Lemma with Varying Measures
In all of the above statements of Fatou's Lemma, the integration was carried out with respect to a single fixed measure μ. Suppose that μn is a sequence of measures on the measurable space (''S'',''Σ'') such that (see Convergence of measures
In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by ''convergence of measures'', consider a sequence of measures μ''n'' on a space, sharing ...
)
:\mu_n(E)\to \mu(E),~\forall E\in \mathcal.
Then, with ''fn'' non-negative integrable functions and ''f'' being their pointwise limit inferior, we have
: \int_S f\,d\mu \leq \liminf_ \int_S f_n\, d\mu_n.
:
Fatou's lemma for conditional expectations
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, by a change of notation, the above versions of Fatou's lemma are applicable to sequences of random variables
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 ...
''X''1, ''X''2, . . . defined on a probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
\scriptstyle(\Omega,\,\mathcal F,\,\mathbb P); the integrals turn into expectations. In addition, there is also a version for conditional expectation
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – give ...
s.
Standard version
Let ''X''1, ''X''2, . . . be a sequence of non-negative random variables on a probability space \scriptstyle(\Omega,\mathcal F,\mathbb P) and let
\scriptstyle \mathcal G\,\subset\,\mathcal F be a sub-σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
. Then
:\mathbb\Bigl \,\mathcal G\Bigrle\liminf_\,\mathbb \mathcal G/math> almost surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
.
Note: Conditional expectation for non-negative random variables is always well defined, finite expectation is not needed.
Proof
Besides a change of notation, the proof is very similar to the one for the standard version of Fatou's lemma above, however the monotone convergence theorem for conditional expectations has to be applied.
Let ''X'' denote the limit inferior of the ''X''''n''. For every natural number ''k'' define pointwise the random variable
:Y_k=\inf_X_n.
Then the sequence ''Y''1, ''Y''2, . . . is increasing and converges pointwise to ''X''.
For ''k'' ≤ ''n'', we have ''Y''''k'' ≤ ''X''''n'', so that
:\mathbb \mathcal Gle\mathbb \mathcal G/math> almost surely
by the monotonicity of conditional expectation, hence
:\mathbb \mathcal Gle\inf_\mathbb \mathcal G/math> almost surely,
because the countable union of the exceptional sets of probability zero is again a null set
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null ...
.
Using the definition of ''X'', its representation as pointwise limit of the ''Y''''k'', the monotone convergence theorem for conditional expectations, the last inequality, and the definition of the limit inferior, it follows that almost surely
:
\begin
\mathbb\Bigl \,\mathcal G\Bigr&=\mathbb \mathcal G=\mathbb\Bigl \,\mathcal G\Bigr=\lim_\mathbb \mathcal G\
&\le\lim_ \inf_\mathbb \mathcal G=\liminf_\,\mathbb \mathcal G
\end
Extension to uniformly integrable negative parts
Let ''X''1, ''X''2, . . . be a sequence of random variables on a probability space \scriptstyle(\Omega,\mathcal F,\mathbb P) and let
\scriptstyle \mathcal G\,\subset\,\mathcal F be a sub-σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
. If the negative parts
:X_n^-:=\max\,\qquad n\in,
are uniformly integrable with respect to the conditional expectation, in the sense that, for ''ε'' > 0 there exists a ''c'' > 0 such that
:\mathbb\bigl \,\mathcal G\bigr\varepsilon,
\qquad\textn\in\mathbb,\,\text,
then
:\mathbb\Bigl \,\mathcal G\Bigrle\liminf_\,\mathbb \mathcal G/math> almost surely.
Note: On the set where
:X:=\liminf_X_n
satisfies
:\mathbb \,\mathcal G\infty,
the left-hand side of the inequality is considered to be plus infinity. The conditional expectation of the limit inferior might not be well defined on this set, because the conditional expectation of the negative part might also be plus infinity.
Proof
Let ''ε'' > 0. Due to uniform integrability with respect to the conditional expectation, there exists a ''c'' > 0 such that
:\mathbb\bigl \,\mathcal G\bigr\varepsilon
\qquad\textn\in\mathbb,\,\text.
Since
:X+c\le\liminf_(X_n+c)^+,
where ''x''+ := max denotes the positive part of a real ''x'', monotonicity of conditional expectation (or the above convention) and the standard version of Fatou's lemma for conditional expectations imply
:\mathbb \,\mathcal Gc
\le\mathbb\Bigl \,\mathcal G\Bigr\le\liminf_\mathbb \,\mathcal G/math> almost surely.
Since
:(X_n+c)^+=(X_n+c)+(X_n+c)^-\le X_n+c+X_n^-1_,
we have
:\mathbb \,\mathcal G\le\mathbb \,\mathcal Gc+\varepsilon almost surely,
hence
:\mathbb \,\mathcal Gle
\liminf_\mathbb \,\mathcal G\varepsilon almost surely.
This implies the assertion.
References
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{{Measure theory
Inequalities
Lemmas in analysis
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