Does $\int_0^\infty f(x+\theta)g(x) \, dx=0 \ , \ \forall \theta \in \mathbb{R}$ imply $f=0$ almost everywhere if $g$ is smooth and strictly positive?

Question

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an integrable function, and $g:(0,\infty)\rightarrow(0,\infty)$ a smooth, strictly positive function.

If

$$\int_0^\infty f(x+\theta)g(x)\,dx=0\qquad\forall\theta\in\mathbb{R}$$

Does that imply that $f=0$ almost everywhere?

This problem is part of a statistics problem in which I would like to prove that if

$$\int_0^\infty f(x+3\theta)ne^{-nx} \, dx=0$$

For all $\theta\in\mathbb{R}$, then one should have $f=0$ almost everywhere; you can find the main question here, in which I propose $X_{(1)}$ as a complete statistic for a given density function - it has been solved, nevertheless I still am interested in a more global and rich treatment of the problem.

Some collegues are suggesting to use fourier analysis and the convolution theorem, but I am not familiar with these techniques and I would like to avoid those methods to solve the problem.

Thanks in advance.