REFERENCE SHEET

Convergence theorems & iterated integrals

Before exchanging a limit or order of integration, identify the theorem whose hypotheses apply.

Before this Lebesgue integration & Lᵖ spaces, Series & uniform convergence

|fₙ|≤g∈L¹, fₙ→f a.e. ⇒ ∫fₙ→∫f

Definitions

Almost-everywhere convergence

Convergence at all points except a measurable null set.

L¹ convergence

∫|fₙ−f|→0. It implies convergence of the integrals and convergence in measure.

σ-finiteness

The space is a countable union of measurable sets of finite measure, a standard hypothesis for product-measure results.

Key theorems

Monotone convergence

If measurable 0≤fₙ↑f pointwise, then ∫fₙ↑∫f, allowing +∞.

Fatou lemma

For nonnegative measurable fₙ, ∫liminf fₙ≤liminf∫fₙ.

Dominated convergence

If measurable fₙ→f almost everywhere and |fₙ|≤g almost everywhere for one integrable g, then f is integrable and fₙ→f in L¹.

Tonelli and Fubini

On σ-finite measure spaces, a nonnegative measurable function on the product has equal iterated and product integrals, possibly infinite (Tonelli). For an absolutely integrable function the same holds with finite values and almost-everywhere-defined sections (Fubini).

Notable applications

Limits of expectations

An integrable envelope permits passing to the limit under expectation.

Multidimensional integration

Nonnegativity or absolute integrability justifies integrating one coordinate at a time.

Work it out

Try before revealing

01 / Diagnose

On (0,1), let fₙ=n·1(0,1/n). Find its pointwise limit and integrals.

Hint

The spike becomes taller as its support shrinks.

Solution

For every fixed x>0, fₙ(x) eventually vanishes, so the limit is zero. But ∫fₙ=1. There is no common integrable dominator; dominated convergence would otherwise force the integrals to tend to zero.

02 / Apply

Show ∫₀¹ xⁿ/(1+x) dx→0.

Hint

Ignore the endpoint x=1, a null set, and find a bound.