Question about integral is equal to zero

Question

Suppose we have the following equation $$ \int_0^\infty {f(x,r)g(x)dx} = 0 \quad {\rm for \, all}\, r\in \mathbb{R} $$ where the function $g(x)$ does not depend on $r$, while $f(x,r)$ is function of $x$ and $r$. Can I conclude that $g(x)= 0$ for all $x\in \mathbb{R}$?

Answer 1

Not necessarily. Thus take any $g \in L^2[0,\infty)$. The orthogonal complement of $g$ is a closed subspace $V$ of $L^2[0,\infty)$ of codimension $1$, and you could take any $f$ so that $f(.,r) \in V$ for all $r$. Examples are very easy to construct.

Answer 2

Depends on what $f$ is. In general, problems of the form "find $g$ given $h(r) = \int_a^b f(x,r) g(r) dr$ are known as Fredholm integral equations of the first kind and are typically not fun.

For example, if $f$ is zero, then $g$ could be anything.