I want to simplify the integral $$I=\int_y \log \left( \int_x f(y) \delta(x-y) dx \right)dy,$$ where $x$, $y$ are real numbers, $f$ is a "nice" real fuction of real argument (eg. exp) and $\delta$ stands for Dirac Delta function.
My idea is to somehow rescale the measure $dy$ by a function $g(x,y)$ such that $$I=\log \left(\int_y \int_x f(y) \delta(x-y) g(x,y) dx dy\right).$$ The question is how to find such a function $g(x,y)$ that would enable to use the sampling property of Dirac Delta function to simplify the integral $I$.
We have that \begin{align*} I &=\int_y \ln \left( \int_x f(y)\, \delta(x-y)\, dx \right)dy\\ &=\int_y \ln \left(f(y) \int_x \, \delta(x-y)\, dx \right)dy\\ &=\int_y \ln \left(f(y) \right)\,dy\\ &=y\ln(f(y))-\int_y y\,\frac{f'(y)}{f(y)}\,dy, \end{align*} using by-parts once. Without knowing what $f(y)$ is, this is likely as far as you can go, if you want exact simplifications and not approximations.