Fatou's lemma

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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 the French mathematician Pierre Fatou (1878 - 1929).

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

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[edit] Standard statement of Fatou's lemma

If f1, f2, . . . is a sequence of non-negative measurable functions defined on a measure space (S,Σ,μ), then

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

Note: On the left-hand side the limit inferior of the fn is taken pointwise. The functions are allowed to attain the value infinity and the integrals may also be infinite.

[edit] Proof

Fatou's lemma is here proved using the monotone convergence theorem (it can be proved directly).

Let f denote the limit inferior of the fn. For every natural number k define pointwise the function

g_k=\inf_{n\ge k}f_n.

Then the sequence g1, g2, . . . is increasing and converges pointwise to f. For k ≤ n, we have gk ≤ fn, so that

\int_S g_k\,d\mu\le\int_S f_n\,d\mu,

hence

\int_S g_k\,d\mu \le\inf_{n\ge k}\int_S f_n\,d\mu.

Using the monotone convergence theorem, the last inequality, and the definition of the limit inferior, it follows that

\int_S \liminf_{n\to\infty} f_n\,d\mu =\lim_{k\to\infty}\int_S g_k\,d\mu \le\lim_{k\to\infty} \inf_{n\ge k}\int_S f_n\,d\mu =\liminf_{n\to\infty} \int_S f_n\,d\mu\,.

[edit] 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 uniform) to the zero function (with zero integral), but every fn has integral one.

[edit] 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 a non-negative integrable function g on S such that fn ≤ g for all n, then

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

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

[edit] Proof

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

[edit] Extensions and variations of Fatou's lemma

[edit] 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\,.

[edit] Proof

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

[edit] 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\,.

[edit] 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.

[edit] Convergence in measure

The last assertion also holds, if the sequence f1, f2, . . . converges in measure to a function f.

[edit] 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.

[edit] 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:

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.

[edit] 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}

[edit] External links

[edit] References

  • H.L. Royden, "Real Analysis", Prentice Hall, 1988.