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In mathematics, Fatou's lemma establishes an
inequality Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...
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 Le ...
of the limit inferior of a sequence of
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 ...
s to the limit inferior of integrals of these functions. The lemma 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 P ...
. Fatou's lemma can be used to prove the Fatou–Lebesgue theorem 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, \operatorname_ denotes the \sigma-algebra of
Borel sets In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...
on ,+\infty/math>. Fatou's lemma remains true if its assumptions hold \mu-almost everywhere. In other words, it is enough that there is a null set N such that the values \ are non-negative for every . To see this, note that the
integrals 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 d ...
appearing in Fatou's lemma are unchanged if we change each function on N.


Proof

Fatou's lemma does ''not'' require the monotone convergence theorem, but the latter can be used to provide a quick proof. A proof directly from the definitions of integrals is given further below. In each case, the proof begins by analyzing the properties of \textstyle g_n(x)=\inf_f_k(x). These satisfy: # the sequence \_n is
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
non-decreasing at any and # g_n\leq f_n, \forall n \in \N. Since f(x) =\liminf_ f_n(x) = \lim_ \inf_ f_k(x)=\lim_, we immediately see that is measurable.


Via the Monotone Convergence Theorem

Moreover, :\int_X f\,d\mu=\int_X\lim_g_n\,d\mu By the Monotone Convergence Theorem and property (1), the limit and integral may be interchanged: :\begin \int_X f\,d\mu&=\lim_\int_X g_n\,d\mu\\ &=\liminf_\int_X g_n\,d\mu\\ &\leq \liminf_\int_X f_n\,d\mu, \end where the last step used property (2).


From "first principles"

To demonstrate that the monotone convergence theorem is not "hidden", the proof below does not use any properties of Lebesgue integral except those established here. Denote by \operatorname(f) the set of
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(\mathcal, \operatorname_)-measurable functions s:X\to Borel σ-algebra and the Lebesgue measure. * Example for a probability space: Let S=[0,1] denote the unit interval. For every natural number n define :: f_n(x)=\beginn&\textx\in (0,1/n),\\ 0&\text \end * Example with uniform convergence: Let S denote the set of all real numbers. Define :: f_n(x)=\begin\frac1n&\textx\in ,n\\ 0&\text \end These sequences (f_n)_ converge on S pointwise (respectively uniformly) to the zero function (with zero integral), but every f_n has integral one.


The role of non-negativity

A suitable assumption concerning the negative parts of the sequence ''f''1, ''f''2, . . . of functions is necessary for Fatou's lemma, as the following example shows. Let ''S'' denote the half line ,∞)_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_[ ,∞)_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.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, sharin ...
) :\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 ''X''1, ''X''2, . . . defined on a probability space \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 – given ...
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. 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. 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. 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

* * * {{Measure theory Inequalities Lemmas in analysis Theorems in measure theory Real analysis Articles containing proofs