CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29

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A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions

Statement

Let $U, V \subseteq R^{n}$ be open and let $T : U \to V$ be a $C^{1}$ diffeomorphism. If $f : V \to C$ belongs to $L^{1} (λ_{n})$ , then

\int_{V} f (y) d λ_{n} (y) = \int_{U} f (T (x)) ∣ \det D T (x) ∣ d λ_{n} (x) .

Facts & Assumptions

Given: Open sets $U, V \subseteq R^{n}$ , a $C^{1}$ diffeomorphism $T : U \to V$ , and a function $f \in L^{1} (λ_{n})$ .

[L1]

The change-of-variables formula holds for nonnegative measurable functions. (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions)

[L2]

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

Proof

technique · direct

1.1

Write $f = u - v + i (p - q)$ , where $u, v, p, q \geq 0$ are the positive and negative parts of the real and imaginary parts of $f$ . Since $f \in L^{1}$ , all four functions are integrable and nonnegative.

L1L2

2.1

Apply [L1] to $u, v, p, q$ and recombine the four resulting equalities by [L2]. This yields the stated formula for $f$ .

step 1.1L1L2∎

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Sources

Gerald B. Folland, Real Analysis, 2nd ed., Theorem 2.47 (standard reference, not scraped)