Proving some sup over integrals is finite (could we exchange $\sup$ and $\int$?). Could someone help me to prove that $$I=\sup_x \int_{|x-y|<1}|q(y)|^2 \omega(x-y) \, dy <\infty$$ where $x,y\in \mathbb{R}^n$, $q(y)$ is bounded for all $y$ and $\omega(x-y)$ is a function given by: $\mathbf{i})$ negative powers of $|x-y|$, e.g., $|x-y|^{-1}$ or $\mathbf{ii})$ $\omega(x-y)=\log |x-y|$ or $\mathbf{iii}$ $\omega(x-y)=1$?
What about to exchange supremum and integral?

It's definitely less than infinity, its producing a real number. So therefore its definitely true. However, if it was something like $$I=\sup_x \int_{|x-y|<1}{|q(y)|^2 \omega(x-y) \, dy >\infty}$$ it'd be false.
Also, $\lim_{n\rightarrow \infty} f_n = \lim_{n\rightarrow \infty} \inf (f_n)  = \lim_{n\rightarrow \infty} \sup (f_n) = f$
Its LIMITING to infinity, therefore it's not necessarily infinity. :)
