The following is an exercise 6.9 in Rudin's Real and Complex Analysis:
Suppose that $\{ g_n \}$ is a sequence of positive continuous functions on $I=[0,1]$, that $\mu$ is a positive Borel measure on $I$, and that
(i) lim$_{n\to \infty}$ $g_n (x) = 0$ a.e. [m],
(ii) $\int_I g_n dm = 1$ for all $n$,
(iii) lim$_{n\to \infty}$ $\int_I fg_n dm = \int_I f d\mu$ for every $f\in C(I)$.
Does it follow that $\mu \perp m$?
- I think that the answer is positive. $\{g_n\}$ seems to be something similar to good kernel. I tried to use Egoroff's theorem and then derive something useful, but couldn't. Would you please give me some help?