Suppose $\xi$ a random variable defined on $\mathbb{R}$ with sigma-algebra $\mathcal{F}$. Let $\mathcal{P}$ denote the set of all probability measures on $(\mathbb{R},\mathcal{F})$. Is $\{P\in \mathcal{P}:\mathbf{E}_P[\xi]\leq 0,\mathbf{E}_P[\xi^2]\leq 1\}$ closed ? You can assume weak convergence
2026-04-07 02:51:13.1775530273
Is the set $\{P\in \mathcal{P}:\mathbf{E}_P[\xi]\leq 0,\mathbf{E}_P[\xi^2]\leq 1\}$ closed?
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