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Fatou's lemma

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Fatou's lemma

In mathematics, Fatou's lemma establishes an inequality relating the integral (in the sense of Lebesgue) of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou.

Fatou's lemma can be used to prove the Fatou–Lebesgue theorem and Lebesgue's dominated convergence theorem.

Contents

  • Standard statement of Fatou's lemma 1
    • Proof 1.1
  • Examples for strict inequality 2
  • The role of non-negativity 3
  • Reverse Fatou lemma 4
    • Proof 4.1
  • Extensions and variations of Fatou's lemma 5
    • Integrable lower bound 5.1
      • Proof 5.1.1
    • Pointwise convergence 5.2
      • Proof 5.2.1
    • Convergence in measure 5.3
      • Proof 5.3.1
    • Fatou's Lemma with Varying Measures 5.4
  • Fatou's lemma for conditional expectations 6
    • Standard version 6.1
      • Proof 6.1.1
    • Extension to uniformly integrable negative parts 6.2
      • Proof 6.2.1
  • References 7
  • External links 8

Standard statement of Fatou's lemma

Let f1, f2, f3, . . . be a sequence of non-negative measurable functions on a measure space (S,Σ,μ). Define the function f : S → [0, ∞] a.e. pointwise limit by

f(s) =\liminf_{n\to\infty} f_n(s),\qquad s\in S.

Then f  is measurable and

\int_S f\,d\mu \le \liminf_{n\to\infty} \int_S f_n\,d\mu\,.

Note: The functions are allowed to attain the value +∞ and the integrals may also be infinite.

Proof

Fatou's lemma may be proved directly as in the first proof presented below, which is an elaboration on the one that can be found in Royden (see the references). The second proof is shorter but uses the monotone convergence theorem – which is usually proved using Fatou's lemma and thus could create a circular argument.

Examples for strict inequality

Equip the space S with the Borel σ-algebra and the Lebesgue measure.

f_n(x)=\begin{cases}n&\text{for }x\in (0,1/n),\\ 0&\text{otherwise.} \end{cases}
f_n(x)=\begin{cases}\frac1n&\text{for }x\in [0,n],\\ 0&\text{otherwise.} \end{cases}

These sequences (f_n)_{n\in\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 f1, f2, . . . of functions is necessary for Fatou's lemma, as the following example shows. Let S denote the half line [0,∞) with the Borel σ-algebra and the Lebesgue measure. For every natural number n define

f_n(x)=\begin{cases}-\frac1n&\text{for }x\in [n,2n],\\ 0&\text{otherwise.} \end{cases}

This sequence converges uniformly on S to the zero function (with zero integral) and for every x ≥ 0 we even have fn(x) = 0 for all n > x (so for every point x the limit 0 is reached in a finite number of steps). However, every function fn has integral −1, hence the inequality in Fatou's lemma fails.

Reverse Fatou lemma

Let f1, f2, . . . 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 fn ≤ g for all n, then

\limsup_{n\to\infty}\int_S f_n\,d\mu\leq\int_S\limsup_{n\to\infty}f_n\,d\mu.

Note: Here g integrable means that g is measurable and that \textstyle\int_S g\,d\mu<\infty.

Proof

Apply Fatou's lemma to the non-negative sequence given by g – fn.

Extensions and variations of Fatou's lemma

Integrable lower bound

Let f1, f2, . . . be a sequence of 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 fn ≥ −g for all n, then

\int_S \liminf_{n\to\infty} f_n\,d\mu \le \liminf_{n\to\infty} \int_S f_n\,d\mu.\

Proof

Apply Fatou's lemma to the non-negative sequence given by fn + g.

Pointwise convergence

If in the previous setting the sequence f1, f2, . . . converges pointwise to a function f μ-almost everywhere on S, then

\int_S f\,d\mu \le \liminf_{n\to\infty} \int_S f_n\,d\mu\,.

Proof

Note that f has to agree with the limit inferior of the functions fn 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 f1, f2, . . . converges in measure to a function f.

Proof

There exists a subsequence such that

\lim_{k\to\infty} \int_S f_{n_k}\,d\mu=\liminf_{n\to\infty} \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)

\mu_n(E)\to \mu(E),~\forall E\in \Sigma.

Then, with fn non-negative integrable functions and f being their pointwise limit inferior, we have

\int_S f\,d\mu \leq \liminf_{n\to \infty} \int_S f_n\, d\mu_n.

Fatou's lemma for conditional expectations

In probability theory, by a change of notation, the above versions of Fatou's lemma are applicable to sequences of random variables X1, X2, . . . 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 expectations.

Standard version

Let X1, X2, . . . 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{E}\Bigl[\liminf_{n\to\infty}X_n\,\Big|\,\mathcal G\Bigr]\le\liminf_{n\to\infty}\,\mathbb{E}[X_n|\mathcal G]   almost surely.

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 Xn. For every natural number k define pointwise the random variable

Y_k=\inf_{n\ge k}X_n.

Then the sequence Y1, Y2, . . . is increasing and converges pointwise to X. For k ≤ n, we have Yk ≤ Xn, so that

\mathbb{E}[Y_k|\mathcal G]\le\mathbb{E}[X_n|\mathcal G]   almost surely

by the monotonicity of conditional expectation, hence

\mathbb{E}[Y_k|\mathcal G]\le\inf_{n\ge k}\mathbb{E}[X_n|\mathcal G]   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 Yk, the monotone convergence theorem for conditional expectations, the last inequality, and the definition of the limit inferior, it follows that almost surely

\begin{align} \mathbb{E}\Bigl[\liminf_{n\to\infty}X_n\,\Big|\,\mathcal G\Bigr] &=\mathbb{E}[X|\mathcal G] =\mathbb{E}\Bigl[\lim_{k\to\infty}Y_k\,\Big|\,\mathcal G\Bigr] =\lim_{k\to\infty}\mathbb{E}[Y_k|\mathcal G]\\ &\le\lim_{k\to\infty} \inf_{n\ge k}\mathbb{E}[X_n|\mathcal G] =\liminf_{n\to\infty}\,\mathbb{E}[X_n|\mathcal G]. \end{align}

Extension to uniformly integrable negative parts

Let X1, X2, . . . 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\{-X_n,0\},\qquad n\in{\mathbb N},

are uniformly integrable with respect to the conditional expectation, in the sense that, for ε > 0 there exists a c > 0 such that

\mathbb{E}\bigl[X_n^-1_{\{X_n^->c\}}\,|\,\mathcal G\bigr]<\varepsilon, \qquad\text{for all }n\in\mathbb{N},\,\text{almost surely},

then

\mathbb{E}\Bigl[\liminf_{n\to\infty}X_n\,\Big|\,\mathcal G\Bigr]\le\liminf_{n\to\infty}\,\mathbb{E}[X_n|\mathcal G]   almost surely.

Note: On the set where

X:=\liminf_{n\to\infty}X_n

satisfies

\mathbb{E}[\max\{X,0\}\,|\,\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{E}\bigl[X_n^-1_{\{X_n^->c\}}\,|\,\mathcal G\bigr]<\varepsilon \qquad\text{for all }n\in\mathbb{N},\,\text{almost surely}.

Since

X+c\le\liminf_{n\to\infty}(X_n+c)^+,

where x+ := max{x,0} 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{E}[X\,|\,\mathcal G]+c \le\mathbb{E}\Bigl[\liminf_{n\to\infty}(X_n+c)^+\,\Big|\,\mathcal G\Bigr] \le\liminf_{n\to\infty}\mathbb{E}[(X_n+c)^+\,|\,\mathcal G]   almost surely.

Since

(X_n+c)^+=(X_n+c)+(X_n+c)^-\le X_n+c+X_n^-1_{\{X_n^->c\}},

we have

\mathbb{E}[(X_n+c)^+\,|\,\mathcal G] \le\mathbb{E}[X_n\,|\,\mathcal G]+c+\varepsilon   almost surely,

hence

\mathbb{E}[X\,|\,\mathcal G]\le \liminf_{n\to\infty}\mathbb{E}[X_n\,|\,\mathcal G]+\varepsilon   almost surely.

This implies the assertion.

References

  • Royden, H.L. (1988). Real Analysis (3rd ed.). 

External links

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