# Move integral inside logarithm

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 $$I$$.

• Is it the Kronecker delta or the Dirac delta? – G. Gare Oct 7 '19 at 14:17
• Dirac delta, thanks for the comment, I fixed the question – Pavel Prochazka Oct 7 '19 at 14:19
• Since the logarithm is concave, you can define $\mu(dx) = \delta(x - y) dx$ and bring the logarithm inside with Jensen's inequality (this gives you a lower bound). Then, Fubini what you get and in the end you get a "nice" lower bound – G. Gare Oct 7 '19 at 14:24
• Note that $\displaystyle\int_x f(y)\,\delta(x-y)\,dx=f(y)\int_x\delta(x-y)\,dx=f(y).$ – Adrian Keister Oct 7 '19 at 14:26

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.