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c.f. Rudin's Real and Complex Analysis (Third Edition 1987) Chapter 6 Q9

Suppose that $\{g_n\}$ is a sequence of positive continuous functions on $I=[0,1]$, $\mu$ is a positive Borel measure on $I$, $m$ is the standard Lebesgue measure, and that

(i) $\lim_{n\to\infty}g_n(x)=0$ a.e. $[m]$

(ii) $\int_Ig_ndm=1$ for all $n$,

(iii) $\lim_{n\to\infty}\int_Ifg_ndm=\int_Ifd\mu$ for every $f\in C(I)$.

Does it follow that the measures $\mu$ and $m$ are mutually singular?

I know that $\mu$ and $m$ are mutually singular if they are concentrated in different disjoint sets, but how do I connect that with the 3 properties above? I will appreciate if someone can help me with the proof or counter example.

2 Answers 2