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Integrating against a density agrees with integrating the product

Statement

Let $f, g : X \to [0, + \infty]$ be measurable. Then $\int g d (f d μ) = \int g f d μ .$

Facts & Assumptions

Given: Nonnegative measurable functions $f$ and $g$ .

[L1]

The density measure is defined by $(f d μ) (A) = \int_{A} f d μ$ (The measure with density $f$ relative to $μ$ ).

[L2]

The nonnegative integral is additive on measurable sets (Additivity of the nonnegative Lebesgue integral).

[L3]

Nonnegative measurable functions admit increasing simple approximations, and products with simple functions are measurable by finite sums of indicator products (Every nonnegative measurable function is the increasing limit of simple measurable functions, Closure properties of measurable functions used by the integral).

[L4]

Monotone convergence holds for the nonnegative integral (Monotone convergence for the integral).

[L5]

The nonnegative integral is homogeneous, and on simple functions it agrees with the simple integral for any measure. (Monotonicity and nonnegative homogeneity of the nonnegative integral, The nonnegative integral agrees with the simple integral on simple functions)

Proof

technique · direct

1.1L1L2L5givenalgebra

Suppose first that $g = \sum_{j = 1}^{m} c_{j} χ_{E_{j}}$ is simple with pairwise disjoint measurable $E_{j}$ . By [L5] for the measure $f d μ$ and then [L1], $\int g d (f d μ) = \sum_{j} c_{j} (f d μ) (E_{j}) = \sum_{j} c_{j} \int_{E_{j}} f d μ$ . Also $g f = \sum_{j} c_{j} f χ_{E_{j}}$ has pairwise disjoint summand supports, so [L2] and [L5] give $\int g f d μ = \sum_{j} c_{j} \int_{E_{j}} f d μ$ . Hence $\int g d (f d μ) = \int g f d μ$ .

2.1step 1.1L3L4∎

For general measurable $g \geq 0$ , choose simple $g_{n} ↑ g$ by [L3]. Then $g_{n} f ↑ g f$ pointwise (using $0 \cdot \infty = 0$ ). Applying [L4] to both measures and step 1.1 to each $g_{n}$ yields $\int g d (f d μ) = \lim_{n} \int g_{n} d (f d μ) = \lim_{n} \int g_{n} f d μ = \int g f d μ$ .

Depends on

Used by

Brownian paths have infinite total variation Corollary
A homogeneous quotient without invariant measure Counterexample
Feller negligibility cannot be removed from the converse Counterexample
Infinite variance can defeat square-root-n CLT scaling Counterexample
Pointwise limit discontinuous at zero signals mass escape Counterexample
Spectral multiplicity function in the separable case Definition
A Brownian hitting time has infinite mean Example
Cauchy law and its characteristic function Example
Characteristic function of a gaussian law Example
Characteristic function of the uniform law Example
CLT for sums of uniform random variables Example
Conditional density of a bivariate normal law Example
Density inversion for a triangular characteristic function Example
Independent sums via characteristic functions Example
The density 2x on [0,1] is the Radon-Nikodym derivative of its density measure Example
Characteristic function of a normal law Lemma
Dentable average ranges give vector-measure densities Lemma
Gaussian even moments for Brownian increments Lemma
Haar change of variables under inversion Lemma
The Brownian kernels form a semigroup Lemma
Unitary intertwiners preserve direct-integral fiber dimension Lemma
Conditional density formula Theorem
Marcinkiewicz interpolation from weak (1,1) and strong (∞,∞) Theorem
Poisson extension is an Lp contraction and converges in finite Lp Theorem
Radon-Nikodym derivatives satisfy the chain rule along nu << mu << lambda Theorem
Unitary equivalence classified by measure class and multiplicity Theorem

Dependency tree · two levels

21 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

Gerald B. Folland, Real Analysis, 2nd ed., §2.2 (standard reference, not scraped)