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Fatou's Lemma

Fubini's Theorem

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Fatou's Lemma

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Statement of Fatou's Lemma

Fatou's Lemma states that for any sequence of non-negative measurable functions $f_n$ , the following inequality holds:

\int \liminf_{n \to \infty} f_n d\mu \leq \liminf_{n \to \infty} \int f_n d\mu

Essentially, it's a statement about the limit of integrals being greater than or equal to the integral of the limits.

Fatou's Lemma with Dominated Convergence Theorem

The Dominated Convergence Theorem can follow from Fatou's Lemma. If $f_n$ converges to $f$ a.e. and is dominated by an integrable function $g$ , then Fatou's Lemma helps to show that

\int f d\mu = \lim_{n \to \infty} \int f_n d\mu

. This is more powerful than Fatou's Lemma as it provides an equality rather than an inequality.

Connection to Lebesgue Integration

Fatou's Lemma is integral within the Lebesgue integration framework as it validates that integration is well-defined in the limit under certain conditions. As an example, consider a sequence of simple functions that approximate a non-negative measurable function; Fatou's Lemma assures that we can interchange limit and integration for this sequence.

Fatou's Lemma in Probability Theory

In the context of probability theory, Fatou's Lemma is often used to show that if $X_n$ are non-negative random variables, then:

\mathbb{E}(\liminf X_n) \leq \liminf \mathbb{E}(X_n)

Here,

\mathbb{E}

denotes the expected value, which is essentially an integral with respect to the probability measure.

Need for Non-negative Functions

Fatou's Lemma requires the functions in the sequence to be non-negative because the lemma relies on the Monotone Convergence Theorem, which in turn is only valid for non-negative functions. If we considered a sequence with negative parts, the lemma might fail as integrals might not be well-defined (e.g., result in $\infty - \infty$ ).

Counterexample Where Fatou's Lemma Does Not Apply

Consider the sequence of functions $f_n(x) = (-1)^n \frac{1}{n} \chi_{[0,1]}(x)$ , where $\chi$ is the characteristic function. Here, $\liminf_{n\to\infty} f_n(x) = 0$ , but

\liminf_{n \to \infty} \int f_n d\mu = 0 \not\leq \int \liminf_{n \to \infty} f_n d\mu = 0

This does not contradict Fatou's Lemma because the functions

f_n

are not non-negative, hence Fatou's Lemma does not apply.

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