I'm trying to solve the following problem.

Let $f$ be an integrable function in (0,1). Suppose that

${\int}_{0}^{1}fg\ge 0$

for any non negative, continuous $g:(0,1)\to \mathbb{R}$. Prove that $f\ge 0$ a.e. in (0,1).

I'm a little unsure on what it is that I must prove in order to conclude that $f\ge 0$. I tried to show that ${\int}_{0}^{1}{f}^{2}\ge 0$ but I couldn't get very far.

I'm seeking hints on how to solve this. Thanks.

Let $f$ be an integrable function in (0,1). Suppose that

${\int}_{0}^{1}fg\ge 0$

for any non negative, continuous $g:(0,1)\to \mathbb{R}$. Prove that $f\ge 0$ a.e. in (0,1).

I'm a little unsure on what it is that I must prove in order to conclude that $f\ge 0$. I tried to show that ${\int}_{0}^{1}{f}^{2}\ge 0$ but I couldn't get very far.

I'm seeking hints on how to solve this. Thanks.