All Questions

Let $f$ be a nonnegative function on the $d$-dimensional torus $\mathbb{T}^d$, which you can take to be smooth. Let $\bar{f}:=\int_{\mathbb{T}^d}fdx$. I am interested in whether the following ...

Let $u\leq v$ be two locally bounded subharmonic functions in a domain in $\mathbb{R}^n$. Assume that $u=v$ on a dense subset. Is it true that $u=v$ everywhere?

I would like to know some examples of the following polar sets (if they exist): a non trivial uncountable polar set in $\mathbb{R}^{2}$; a polar set in $\mathbb{R}^{2}$ contained in $\mathbb{R}$ with ...