Lebesgue Integration - Key Proof

ProofProof of Fatou's Lemma

We prove Fatou's Lemma: For non-negative measurable functions $\{f_n\}$ , $\int_X \liminf_{n \to \infty} f_n \, d\mu \leq \liminf_{n \to \infty} \int_X f_n \, d\mu$

Proof: For each $k \in \mathbb{N}$ , define: $g_k(x) = \inf_{n \geq k} f_n(x)$

Then $g_k$ is measurable (as the infimum of measurable functions) and: $g_1(x) \leq g_2(x) \leq g_3(x) \leq \cdots$

Moreover: $\lim_{k \to \infty} g_k(x) = \liminf_{n \to \infty} f_n(x)$

Step 1: Since $g_k(x) \leq f_n(x)$ for all $n \geq k$ , we have: $\int_X g_k \, d\mu \leq \int_X f_n \, d\mu \text{ for all } n \geq k$

Taking the infimum over $n \geq k$ : $\int_X g_k \, d\mu \leq \inf_{n \geq k} \int_X f_n \, d\mu$

Step 2: Since $\{g_k\}$ is an increasing sequence of non-negative functions, we can apply the Monotone Convergence Theorem: $\int_X \liminf_{n \to \infty} f_n \, d\mu = \int_X \lim_{k \to \infty} g_k \, d\mu = \lim_{k \to \infty} \int_X g_k \, d\mu$

Step 3: From Step 1: $\lim_{k \to \infty} \int_X g_k \, d\mu \leq \lim_{k \to \infty} \inf_{n \geq k} \int_X f_n \, d\mu = \liminf_{n \to \infty} \int_X f_n \, d\mu$

Combining Steps 2 and 3 gives the result.

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Remark

Fatou's Lemma is fundamental because:

Strict inequality possible: The inequality can be strict. For example, let $f_n = \chi_{[n, n+1]}$ on $\mathbb{R}$ . Then $f_n \to 0$ pointwise, so: $\int_\mathbb{R} \liminf f_n \, d\lambda = 0 < 1 = \liminf \int_\mathbb{R} f_n \, d\lambda$
Proof ingredient for DCT: Fatou's Lemma is used to prove the Dominated Convergence Theorem by applying it to $g \pm f_n$ where $g$ is the dominating function.
Reverse Fatou: If $f_n \leq g$ for integrable $g$ , then: $\limsup_{n \to \infty} \int_X f_n \, d\mu \leq \int_X \limsup_{n \to \infty} f_n \, d\mu$

This follows from applying Fatou's Lemma to $g - f_n$ .

The key insight is that MCT provides a way to handle monotone sequences exactly, while Fatou's Lemma handles general sequences by considering their "monotone regularization" through the $\inf$ operation.