While doing my study on the boundary-crossing time of a stochastic process, I happened to deal with the following question which is somehow related to Fredholm theory.

Question : Suppose $K$ is continuous non-negative function on $[0,T]^2$ such that $K(t,s)= 0 $ if and only if $s \ge t$. Prove or disprove the following statement:

"If $f$ is integrable measurable function on $[0,T]$, then $ \int_{[0,T]} K(t,s)f(s)ds= 0 \forall t \iff f= 0 $ a.s"

Thought : This statement is reasonable because for each $t$, $K(t, \cdot)$ can mimic the role of the indicator function $\mathbb{1}_{[0,s]}$ .

I'm essentialy looking for answer helping me proceed with that question, for example, an approach, and a condition to be added in order to make the question more agreeable,etc. Nevertheless, no matter how your help is, I always deeply appreciate it.

Thank you.

While doing my study on the boundary-crossing time of a stochastic process, I happened to deal with the following question which is somehow related to Fredholm theory.

Question : Suppose $K$ is continuous non-negative function on $[0,T]^2$ such that $K(t,s)= 0 $ if and only if $s \ge t$. Prove or disprove the following statement:

"If $f$ is integrable function on $[0,T]$, then $ \int_{[0,T]} K(t,s)f(s)ds= 0 \hspace{3mm} \forall t \iff f= 0 $ a.s"

Thought : This statement is reasonable because for each $t$, $K(t, \cdot)$ can mimic the role of the indicator function $\mathbb{1}_{[0,t]}$ .

I'm essentially looking for an answer helping me proceed with that question, for example, an approach, or a condition to be added in order to make the question more agreeable,etc. Nevertheless, no matter how your help is, I always deeply appreciate it.

Thank you.