Thread: Lebesgue measure problem

Hi,

I have the following problem:

Let m be the Lebesgue measure on R and let v be a measure on the Lebesgue sigma-algebra such that:
(i) v is absolutely cont. with respect to m.
(ii) v({x}+A)=v(A) for all x in R and A in the Lebesgue sigma-algebra.
Show: v=c*m for some constant c in R.

Really all I can see is that (and this is given by assumption (i)) the statement holds for all E such that m(E)=0. Other then that, nada. Any hints?

Apply Radon-Nikodym theorem: it will give you an integrable function such that for all Lebesgue-measurable, . Now, fix , and put . h is integrable and for all Lebesgue-measurable, . It shows that is constant.