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Let $\mu$ be a Borel measure on $[0, 1]$. Assume that

a) $\mu$ and Lebesgue measure $m$ are mutually singular.

b) $\mu([0, t])$ depends continuously on $t$.

c) $f\in L^1(\mu)$ for any function $f : [0, 1] \rightarrow \mathbb{R}$, with $f\in L^1(m)$. (Note that $f$ has a finite value at every point.)

How can I show that $\mu=0$?

Thank you!

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Maybe Lebesgue decomposition of measures can be useful. – Davide Giraudo Feb 1 '13 at 21:05

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