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No. Let the compact metric space be $[0,1]$ with the standard topology and define $$ f_\alpha(x) =\alpha\max(1-\alpha \vert x\vert,0) $$ for $\alpha\in[0,1]$. This satisfies the properties asked for, …

As mentioned in the question, the inequality $$ \begin{align}\int(f+g)\, d\nu\ge\int f\,d\nu+\int g\,d\nu&&{\rm(1)}\end{align} $$ follows easily from the definition of the integral $\int f\,d\nu$ (as …

If $\mathbb{E}\left[\lVert[ X\rVert^d\right]$ is finite then the integral in the question is necessarily finite. As mentioned, this holds whenever $\pi$ is radially decreasing. However, in the general …

The statement in the question is not true. Given any two enumerable and dense sets in open intervals of the reals, there is a (complex) analytic1 function giving a bijection between them. See the foll …

A while ago, I came across the following problem, which I was not able to resolve one way or the other. Let $f,g\colon\mathbb{R}^2\to\mathbb{R}$ be continuous functions such that $f(t,x)$ and $g(t …

I still have no idea what the answer to this question is. However, it is possible to attack the problem in several different ways, and there are various different (but logically equivalent) ways of st …

Some time ago I spent a lot of effort trying to show that the semimartingale property is preserved by certain functions. Specifically, that a convex function of a semimartingale and decreasing functio …

Yes, it is true that $R$ must be the field of real numbers. As $R$ is an ordered field, it is naturally an extension $\mathbb{Q}\hookrightarrow R$. We can prove the following two properties, which ch …