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Dominated convergence

Statement

Let $f$ and $(f_{n})$ be measurable complex-valued functions such that $f_{n} \to f$ almost everywhere and $∣ f_{n} ∣ \leq g$ almost everywhere for a single nonnegative measurable function $g$ with $\int g d μ < + \infty$ . Then $f \in L^{1} (μ)$ , $\int ∣ f_{n} - f ∣ d μ ⟶ 0,$ and hence $\int f_{n} d μ ⟶ \int f d μ .$

Facts & Assumptions

Given: Measurable complex-valued functions $f, f_{n}$ with $f_{n} \to f$ almost everywhere and $∣ f_{n} ∣ \leq g$ almost everywhere for one nonnegative measurable function $g$ of finite integral.

[L1]

Reverse Fatou's lemma holds under an integrable majorant (Reverse Fatou's lemma under an integrable majorant).

[L2]

The integral is linear on $L^{1} (μ)$ (The Lebesgue integral is linear on $L^{1} (μ)$ ).

[L3]

The integral triangle inequality holds on $L^{1} (μ)$ (The modulus of an integral is bounded by the integral of the modulus).

[L4]

Real and complex integrability are defined in Integrable real and complex functions, and their integrals.

[L5]

The nonnegative integral is additive, and a nonnegative integral over a null set vanishes (Additivity of the nonnegative Lebesgue integral, A nonnegative integral over a null set vanishes).

[L6]

A nonnegative measurable function with finite integral is finite almost everywhere (A nonnegative measurable function with finite integral is finite almost everywhere).

[L7]

The nonnegative integral is monotone and homogeneous (Monotonicity and nonnegative homogeneity of the nonnegative integral).

Proof

technique · direct

1.1L4L5L6L7given

Let $N_{0}$ be a measurable null set outside which $f_{n} (x) \to f (x)$ and $∣ f_{n} (x) ∣ \leq g (x)$ for every $n$ , and let $N_{\infty} : = {g = \infty}$ . By [L6], the set $N_{\infty}$ is null. Put $N : = N_{0} \cup N_{\infty}$ , $E : = X ∖ N$ , $\tilde{g} : = g χ_{E}, h_{n} : = ∣ f_{n} - f ∣ χ_{E} .$ Then $h_{n} \to 0$ pointwise, $0 \leq h_{n} \leq 2 \tilde{g}$ , and $\tilde{g}$ is nonnegative, measurable, and finite everywhere. Also $∣ f ∣ \leq ∣ f ∣ χ_{N} + \tilde{g} .$ By [L5] and [L7], $\int ∣ f ∣ d μ \leq \int ∣ f ∣ χ_{N} d μ + \int \tilde{g} d μ = 0 + \int \tilde{g} d μ \leq \int g d μ < + \infty,$ so $f \in L^{1} (μ)$ by [L4]. The same null-set and domination argument gives $\int ∣ f_{n} ∣ d μ \leq \int g d μ < \infty$ for each $n$ , so every $f_{n}$ also belongs to $L^{1} (μ)$ .

2.1step 1.1L1

The functions $h_{n}$ are nonnegative, converge pointwise to $0$ , and are dominated by the finite everywhere majorant $2 \tilde{g}$ . Applying [L1] therefore gives ${lim sup}_{n} \int h_{n} d μ \leq \int 0 d μ = 0.$ Hence $\int h_{n} d μ \to 0$ .

3.1step 1.1step 2.1L2L3L5∎

Because $∣ f_{n} - f ∣ χ_{N}$ is supported on the null set $N$ , [L5] gives $\int ∣ f_{n} - f ∣ d μ = \int h_{n} d μ + \int ∣ f_{n} - f ∣ χ_{N} d μ = \int h_{n} d μ ⟶ 0.$ Therefore, by [L2] and [L3], $∣ \int f_{n} d μ - \int f d μ ∣ = ∣ \int (f_{n} - f) d μ ∣ \leq \int ∣ f_{n} - f ∣ d μ,$ and the right-hand side tends to $0$ .

Depends on

Used by

Bounded convergence on a finite measure space Corollary
Compactly supported smooth functions are dense in W^k,p(Rⁿ) Corollary
Compatible extensions from the finite simple core Corollary
Density inversion from an integrable characteristic function Corollary
Expected duration of symmetric gambler's ruin Corollary
Finite propagation for scalar conservation laws Corollary
Heat-semigroup martingales Corollary
Interior gradient bound for Poisson solutions Corollary
Poincare-Wirtinger on bounded convex domains by the direct pairwise argument Corollary
Positive, negative, and truncated Sobolev functions Corollary
Schwartz convolution and product laws Corollary
Stationary irreducible Markov shift is ergodic Corollary
The constructed classical solutions are locally determined by the Cauchy data Corollary
Uniqueness, comparison and order preservation of entropy solutions Corollary
Wald first equation under integrable stopping Corollary
Weak differentiation has a closed graph on its natural domains Corollary
Weighted-integrable Hᵖ functions have vanishing moments in the atomic range Corollary
A mild heat solution need not be classical at the initial time Counterexample
A non-Dini continuous Poisson source can destroy the continuity of the second derivatives Counterexample
A nonintegrable observable with divergent ergodic averages Counterexample
A nonzero boundary value creates a zero-extension jump Counterexample
A strongly continuous unitary group need not be norm continuous Counterexample
A symmetric closed operator that is not self-adjoint Counterexample
An unbounded stopped exponential martingale needs uniform integrability Counterexample
Expanding bumps lose tightness Counterexample
Fatou can be strict and domination can fail simultaneously Counterexample
Hilbert transform is not strong type (1,1) Counterexample
Infinite variance can defeat square-root-n CLT scaling Counterexample
Strong type (1,1) fails for the Hilbert transform Counterexample
The critical Sobolev embedding is not compact Counterexample
The regular representation of R is not a Hilbert direct sum of irreducibles Counterexample
The unitary dual need not be Hausdorff Counterexample
There is no universal Riemann–Lebesgue decay rate Counterexample
Integral of a measurable function against a projection-valued measure Definition
Maximal truncated singular integrals Definition
Relative compactness with respect to an operator Definition
The complex-time heat kernel on a proper sector Definition
The Duhamel heat potential Definition
The Littlewood-Paley square function Definition
The Lusin area function for a fixed admissible kernel and aperture Definition

…and 226 more results.

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John K. Hunter, Measure Theory Notes, Theorem 4.24 (standard reference, not scraped)
Gerald B. Folland, Real Analysis, 2nd ed., Theorem 2.24 (standard reference, not scraped)