Measure Theory 7 | Monotone Convergence Theorem (and more) [dark version]

TL;DR

The Lebesgue integral for nonnegative measurable functions is defined as a supremum over step functions that lie below the function.

Briefing Cornell Notes

Briefing

Monotone convergence is the headline result: for a nonnegative measurable sequence (f_n) that increases pointwise to a limit function f, the Lebesgue integral satisfies

lim_{n→∞} ∫ f_n dμ = ∫ f dμ.

That “push the limit inside the integral” move matters because it turns a hard limit-of-functions problem into a tractable limit-of-numbers problem—one of the main reasons Lebesgue integration is so powerful compared with Riemann integration.

Before stating the theorem, the discussion builds the technical machinery needed to justify it. Everything starts with the Lebesgue integral for nonnegative measurable functions, defined via a supremum over step functions lying below the function. Concretely, ∫ f dμ is a number in [0,∞], obtained as the supremum of ∫ h dμ over step functions h with 0 ≤ h ≤ f. From this definition, three key properties are established.

First, the integral depends only on the function’s values almost everywhere: if two nonnegative measurable functions f and g agree except on a set of μ-measure zero, then their integrals are equal. The intuition is that measure-zero changes are invisible to the integral. A proof is sketched using a simple function h and constructing a modified version h~ that differs from h only on a μ-null set. Because the integral of a nonnegative function over a null set contributes nothing, the supremum defining the integral remains unchanged.

Second, monotonicity holds: if f ≤ g almost everywhere, then ∫ f dμ ≤ ∫ g dμ. The argument again leans on the “supremum of step functions below” definition. Step functions that sit below f also sit below g on the relevant full-measure set, so the supremum for f cannot exceed the supremum for g.

Third, for nonnegative functions, “integral zero forces the function to be zero almost everywhere.” Since there is no cancellation between positive and negative parts (everything is ≥ 0), an integral of 0 can only happen if the function vanishes except possibly on a μ-null set.

With these properties in place, the monotone convergence theorem is formally stated. The setting is a measure space (X, Σ, μ) and a sequence of nonnegative measurable functions f_n: X → [0,∞]. The sequence increases pointwise almost everywhere: f_1 ≤ f_2 ≤ … with the inequality failing only on a μ-null set. The pointwise limit exists almost everywhere and equals f. Under these conditions, the theorem guarantees that the limit of the integrals equals the integral of the limit.

The proof of the monotone convergence theorem is deferred to the next installment, but the groundwork makes clear why it will work: the integral’s definition via step-function suprema, together with almost-everywhere invariance and monotonicity, is exactly the structure needed to control what happens as n grows.

Cornell Notes

For nonnegative measurable functions, the Lebesgue integral is defined as a supremum over step functions that lie below the target function. That definition yields three crucial facts: (1) changing a function on a μ-null set does not change its integral, (2) if f ≤ g almost everywhere then ∫f dμ ≤ ∫g dμ, and (3) for nonnegative functions, ∫f dμ = 0 implies f = 0 almost everywhere. These properties set up the Monotone Convergence Theorem: if f_n ≥ 0 increases pointwise almost everywhere to f, then the limit can be moved inside the integral, giving lim_n ∫ f_n dμ = ∫ f dμ. This is a major advantage over Riemann integration because it handles limits of functions cleanly.

Why does the Lebesgue integral ignore changes on sets of measure zero?

Because the integral for nonnegative measurable functions is built from step functions via a supremum. If a function h is modified to h~ only on a μ-null set N, then the integral of the difference contributes nothing: the added term is a times μ(N) = 0. In the proof sketch, h~ matches h on a full-measure set X~ and differs only on X\X~ (a null set), so ∫h~ dμ = ∫h dμ. Since the supremum defining the integral is unaffected by what happens on N, the integral depends only on values almost everywhere.

How does monotonicity (f ≤ g a.e. ⇒ ∫f ≤ ∫g) follow from the “supremum of step functions below” definition?

For ∫f, consider all step functions S that satisfy S ≤ f (on the relevant full-measure set). If f ≤ g almost everywhere, then on that same full-measure set every such S also satisfies S ≤ g. That means the supremum used to define ∫f cannot exceed the supremum used for ∫g. The proof uses a set X~ where f ≤ g holds and then argues that step functions can be restricted to X~ without changing the supremum, because changes outside X~ occur on a μ-null set.

Why does ∫f dμ = 0 force f = 0 almost everywhere for nonnegative functions?

Nonnegativity removes cancellation. Since f ≥ 0, the “area” represented by f cannot be offset by negative contributions. If the integral is 0, the only way the supremum of step-function areas below f can be 0 is if f has no positive mass except possibly on a μ-null set. Thus f must be 0 almost everywhere.

What exactly does “f_n increases to f almost everywhere” mean in the theorem’s hypotheses?

Two almost-everywhere conditions are required. First, the monotone inequality f_1 ≤ f_2 ≤ … holds except on a μ-null set. Second, the pointwise limit lim_n f_n(x) = f(x) holds except on another μ-null set. Taking the intersection of the full-measure sets gives a single set X~ of full measure where both properties hold simultaneously.

What does it mean to “push the limit inside the integral” in Monotone Convergence?

It means replacing a limit of integrals with an integral of a pointwise limit. Under the monotone increase assumption, the theorem guarantees lim_{n→∞} ∫ f_n dμ = ∫ (lim_{n→∞} f_n) dμ = ∫ f dμ. So instead of analyzing the sequence of numbers ∫ f_n dμ directly, one can study the pointwise limit function f and integrate it.

Review Questions

In the definition of the Lebesgue integral for nonnegative functions, what role do step functions play, and why is a supremum used?
Explain how almost-everywhere equality leads to equality of integrals for nonnegative measurable functions.
State the Monotone Convergence Theorem precisely, including the almost-everywhere conditions.

Key Points

1
The Lebesgue integral for nonnegative measurable functions is defined as a supremum over step functions that lie below the function.
2
Changing a nonnegative measurable function on a μ-null set does not change its Lebesgue integral.
3
If f ≤ g almost everywhere for nonnegative measurable functions, then ∫ f dμ ≤ ∫ g dμ (monotonicity).
4
For nonnegative functions, ∫ f dμ = 0 implies f = 0 almost everywhere, since there is no cancellation.
5
Monotone Convergence applies to nonnegative measurable sequences (f_n) that increase pointwise almost everywhere to f.
6
Under those hypotheses, the limit commutes with integration: lim_n ∫ f_n dμ = ∫ f dμ.
7
The proof of Monotone Convergence is postponed, but the earlier properties are the technical foundation needed for it.

Highlights

The integral is constructed from step functions via a supremum, which makes almost-everywhere behavior central.

A μ-null set can be altered arbitrarily without affecting the integral’s value.

Monotonicity of the integral follows directly from the “step functions below” viewpoint.

Monotone Convergence is the key rule for moving limits inside the Lebesgue integral when f_n increases to f.

Topics

Lebesgue Integral
Almost Everywhere
Monotonicity
Monotone Convergence Theorem
Step Functions