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daon
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Let $P\mathbb{R}$ be the space of probability measures on $\mathbb{R}$ (with its Borel sets). Is the function \begin{align*} P\mathbb{R} &\to \mathbb{N}^\infty\\ \mu &\mapsto \#\{ x \in \mathbb{R} \mid \mu\{x\} > 0 \} \end{align*}\begin{align*} P\mathbb{R} &\to \mathbb{N} \cup \{\infty\}\\ \mu &\mapsto \#\{ x \in \mathbb{R} \mid \mu\{x\} > 0 \} \end{align*} measurable?

Let $P\mathbb{R}$ be the space of probability measures on $\mathbb{R}$ (with its Borel sets). Is the function \begin{align*} P\mathbb{R} &\to \mathbb{N}^\infty\\ \mu &\mapsto \#\{ x \in \mathbb{R} \mid \mu\{x\} > 0 \} \end{align*} measurable?

Let $P\mathbb{R}$ be the space of probability measures on $\mathbb{R}$. Is the function \begin{align*} P\mathbb{R} &\to \mathbb{N} \cup \{\infty\}\\ \mu &\mapsto \#\{ x \in \mathbb{R} \mid \mu\{x\} > 0 \} \end{align*} measurable?

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daon
daon
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Number of atoms of a probability measure

Let $P\mathbb{R}$ be the space of probability measures on $\mathbb{R}$ (with its Borel sets). Is the function \begin{align*} P\mathbb{R} &\to \mathbb{N}^\infty\\ \mu &\mapsto \#\{ x \in \mathbb{R} \mid \mu\{x\} > 0 \} \end{align*} measurable?