Measure Theory 16 | Proof of the Substitution Rule for Measure Spaces [dark version]

The substitution rule for measure spaces lets integrals be transferred across a measurable map: integrating a function on Y with respect to the image measure can be replaced by integrating the same function composed with the map on X. The key requirement is a measurable function H: X → Y between measure spaces (X, μ) and (Y, ν), where ν is the image measure H_*μ. Under the mild condition that at least one of the two integrals exists, the rule guarantees equality of the two sides—so long as measurability and integrability assumptions line up.

The proof starts with the simplest measurable functions on Y: characteristic functions χ_C of measurable sets C ⊂ Y. For such functions, the integral on the Y-side is just the measure of C. Because the measure on Y is the image measure H_*μ, that quantity equals μ(H^{-1}(C)). On the X-side, the composed function χ_C∘H takes value 1 exactly when H(x) ∈ C, which is the same as x ∈ H^{-1}(C). Therefore, integrating χ_C∘H over X again produces μ(H^{-1}(C)). This establishes the substitution identity for characteristic functions.

Next comes the extension to simple functions. Any nonnegative simple function can be written as a finite linear combination of characteristic functions, say G = Σ_{i=1}^n λ_i χ_{C_i}. Linearity of the integral allows the substitution rule to be applied term-by-term, moving from the characteristic-function case to the full simple-function case. At this stage, the equality holds for all simple measurable G by combining the already-proved set-level identity with the integral’s linearity.

To reach general nonnegative measurable functions, the proof uses the definition of the (extended) Lebesgue integral via the supremum over simple functions. For a nonnegative measurable G on Y, one considers simple functions G-tilde on Y that lie pointwise below G. Composing with H transfers these simple functions to X: G-tilde∘H becomes a simple function on X and remains bounded above by G∘H. Since the substitution rule already holds for each such simple function, taking the supremum over all admissible G-tilde yields equality of the integrals for G∘H and G with respect to the image measure.

Finally, the argument handles arbitrary measurable G by splitting it into positive and negative parts. The substitution rule applies separately to the nonnegative components, and if one of the two integrals exists (in the extended sense), the other exists as well. The result is a complete proof of the substitution rule: integrating G over Y with respect to H_*μ equals integrating G∘H over X with respect to μ, provided the relevant integrability condition is satisfied.