Hi, friends! I read that, if ##f\in L^1[c,d]## is a Lebesgue summable function on ##[a,b]## and ##g:[a,b]\to[c,d]## is a(adsbygoogle = window.adsbygoogle || []).push({}); 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!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I Substitution in a Lebesgue integral

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads for Substitution Lebesgue integral |
---|

A How can I Prove the following Integral Inequality? |

I Differentiating a particular integral (retarded potential) |

I Differentiation under the integral in retarded potentials |

I Laplacian of retarded potential |

I Limitations of the Lebesgue Integral |

**Physics Forums | Science Articles, Homework Help, Discussion**