Measure Theory 9 | Fatou's Lemma [dark version]

Fatou’s Lemma gives a one-sided way to move “liminf” through an integral for non-negative measurable functions. Instead of asking whether an integral of a pointwise limit equals the limit of integrals, it focuses on the limit inferior: for a sequence (f_n) of non-negative measurable functions on a measure space (X, Σ, μ), Fatou’s Lemma guarantees

∫_X liminf_{n→∞} f_n dμ ≤ liminf_{n→∞} ∫_X f_n dμ.

That inequality is weaker than a full convergence theorem, but it is powerful because it requires almost nothing beyond measurability and non-negativity. The lemma’s generality makes it a workhorse for later results—most notably Fatou’s Lemma is positioned as a stepping stone toward the more familiar “reverse” convergence statement known as Fatou’s lemma’s companion, which leads into the next theorem in the series.

A key part of the argument is understanding what liminf_{n→∞} f_n means as a function. For each x in X, liminf_{n→∞} f_n(x) is defined using the tail behavior of the sequence of numbers f_k(x):

liminf_{n→∞} f_n(x) = lim_{n→∞} (inf_{k≥n} f_k(x)).

Because the infimum over k≥n can be 0 or potentially grow without bound, the construction naturally lives in the extended non-negative reals (allowing ∞). This choice matters for the statement’s strength: including ∞ from the start avoids edge cases where the liminf might otherwise be undefined.

The proof also leans on measurability and monotonicity. The function g(x) := liminf f_n(x) is measurable because it is built from measurable functions using operations that preserve measurability: taking infima over tails and then taking limits. To make the monotone convergence theorem usable, the proof defines auxiliary functions g_n(x) := inf_{k≥n} f_k(x). These g_n form an increasing sequence: shifting the cutoff n to the right can only increase the infimum, so g_1 ≤ g_2 ≤ g_3 ≤ … .

With that setup, the monotone convergence theorem turns the limit of integrals into an equality:

∫_X lim_{n→∞} g_n dμ = lim_{n→∞} ∫_X g_n dμ.

Since g_n(x) is always ≤ f_n(x) (because g_n takes the infimum over a set that includes f_n), one gets g_n ≤ f_n pointwise, and therefore ∫ g_n dμ ≤ ∫ f_n dμ by monotonicity of the integral. Combining these steps yields the desired inequality with liminf on both sides.

In short: Fatou’s Lemma turns tail infima into a measurable increasing sequence, uses monotone convergence to pass a limit through the integral, and then compares those tail infima to the original functions. The result is a robust inequality that often replaces stronger convergence assumptions when only non-negative measurability is available.