Consider a integral equation $$ \begin{aligned} \mathbb{E} \left[ Y|A \right] &=\mathbb{E} \left[ g\left( W \right) |A \right]\\ \int{yp\left( y|a \right) dy}&=\int{g\left( w \right) p\left( w|a \right) dw}\\ \end{aligned} $$ We have $n$ i.i.d samples $(W,A,Y)$ and hope to solve $g(w)$.
My question is: how to check whether the solution $g(w)$ of the integral equation exists? Or I want to verify whether this integral equation has a solution.
I know that the above equation is called the Fredholm integral equation of the first kind in functional analysis. If it is a compact operator and certain conditions are met, the above equation must have a solution. However, I want to use hypothesis testing to test whether he has a solution.
When $g(w)$ is a parametric model, I have an idea to divide the data set into data sets first. Then use the first data set to solve the integral equation (such as using the generalized method of moments), and then use the second sample to test how far away the error $\left\| \mathbb{E} \left[ Y-g\left( W;\theta \right) |A \right] \right\| ^2$ is from zero to assess how well the solution fits the equation. This is a typical goodness-of-fit test if all models are parametric.
I don't know if my idea is reasonable. Or if it is unreasonable, is there a better way to test it? Is there any relevant literature discussing this issue?
Besides, I don't have any idea when $g(w)$ is a non-parametric model. This is a problem of statistical inference from integral equations. Is there any relevant literature discussing this issue?