Function zero almost everywhere if $\int fg=0$

Assume $f$ is an integrable function on $\mathbb{R^n}$. Assume for every bounded continuous function g on $\mathbb{R^n}$, $\int_\mathbb{R^n}fg=0$. Prove $f$ must equal $0$ almost everywhere.

I am really not sure how to do this problem. I have tried simple stuff but that is not currently working.

• hint : This is a simple aplication of the classical Du Bois - Raymond lemma =) . – math student Jul 26 '13 at 15:16
• @LeandroTavares Any ideas other than that? I am studying for a comp exam and this is part of a test bank provided to us. However, we did not discuss this lemma. – Leo Spencer Jul 26 '13 at 15:23

1. Since the characteristic function of a closed set is a pointwise non-increasing limit of continuous functions, we can prove by a monotone convergence argument that $\int_{\mathbb R^n}f\chi_F=0$ for each closed subset $F$ of $\mathbb R^n$.
2. For each $\varepsilon>0$ and Borel set $S$ of finite measure, we can find a closed subset $F$ contained in $S$ such that $\lambda_n(S\setminus F)<\varepsilon$. Hence $\int_{\mathbb R^n}f\chi_S=0$ for each Borel subset of finite measure.
3. This can be extended to each Borel subset. Indeed, if $B$ is a Borel subset of $\mathbb R^n$, then $B\cap [0,N]^n$ is a Borel subset of finite measure. By 2., this implies that $\int_{\mathbb R^n } f\mathbb 1_{ B\cap [0,N]^n}=0$ for each $N$. Then use the dominated convergence theorem for the sequence $\left(f_N\right)_N$ defined by $f_N:= f\mathbb 1_{ B\cap [0,N]^n}$.