- #1

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*differomorphism*(would it be enough for ##g## to be invertible and such that ##g\in C^1[a,b]## and ##g^{-1}\in C^1[a,b]##, then $$\int_\limits{g([a,b])}f(x)\,d\mu_x=\int_\limits{[a,b]}f(g(t))|g'(t)|\,d\mu_t$$where $\mu$ is the linear Lebesgue measure.

I know that the function ##F## defined by $$F(x):=\int_\limits{[c,x]}f(\xi)\,d\mu_{\xi}$$is absolutely continuous, and that the derivative ##\varphi## of an absolutely continuous function ##\Phi:[c,d]\to\mathbb{R}##, which exists almost everywhere on ##[c,d]##, is such that $$\int_\limits{[c,d]}\varphi(\xi) \,d\mu_{\xi}=\Phi(d)-\Phi(c)$$but I cannot use these two facts alone to prove the desired result.

I do see, for ex. for a non-decreasing ##g##, that ##\frac{d}{dt}\int_\limits{[g(a),g(t)]}f(x)\,d\mu_x=F'(g(t))g'(t)## exists and is equal to ##f(g(t))g'(t)## for almost every ##g(t)## (and therefore for almost every ##t##, since I think that this implies that a homeomorphism like ##g## maps null measure sets to null measure sets), but I am not able to derived the desired identity from this.

How can it be proved? I thank you any answerer very much!