Let $X$ be a compact metrizable space, and $m$ be a finite signed, Borel, regular measure on $X$. Assume that for every continuous function $f:X\to [0,\infty)$ one has $\int_Xfdm\ge0$. How to show $m$ is a positive measure?
My attempt: By the Hahn decomposition of the singed measure $m$, we have $X=N\cup P$, s.t. $N$ is negative, $P$ is positive. Suppose $N$ is not with zero measure. Then we consider the simple function $\chi_N$, so that $\int_X\chi_Ndm=m(N)<0$. Since $m$ is a finite measure, we have $\chi_N$ is $L^1$, so we have continuous functions $f_i$ converging to $\chi_N$ in measure. Assume $\int_X\chi_Ndm=-k<0$, then we can find a continuous function $f_n$ with $\int_X|f_n|-|\chi_N|dm\le \int_X|f_n-\chi_N|dm<k/2$, thus for the continuous function $|f_n|$, we have $0\le\int_X|f_n|dm\le \int_X\chi_Ndm+k/2=-k/2<0$, contradiction.
Am I right? Indeed, I am not sure if we have the theorem that in general for an $L^1$ function, we can have a sequence of continuous functions to converge to in measure. on Folland's book, I only see this result on $\mathbb{R}^1$.