I know the Lebesgue measure is translation invariant. However, how do I prove that $\int_{\mathbb{R}}f(x) dm(x)=\int_{\mathbb{R}}f(x-k)dm(x)$

I know in the case of the Riemann integral and pictures, the result is obvious as the area of the graph under the curve isn't changed by shifting.

  • $\begingroup$ A change of variable? $\endgroup$ – астон вілла олоф мэллбэрг Dec 7 '16 at 4:48
  • $\begingroup$ Try to prove that you can change variables using translation, that means Lebesgue measure is invariant under translations $\endgroup$ – Rono Dec 7 '16 at 4:56
  • $\begingroup$ Alright thanks, I will try it out. $\endgroup$ – Matthew Cheung Dec 7 '16 at 4:59

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