# Measurability of the map $x\mapsto \delta_x$

Let $X$ be a real-valued random variable defined on a measurable space $(\Omega, \mathscr{F})$. Let also $(\mathcal{P}(\Omega), \mathscr{P})$ be the set of probability measure on $\Omega$; here the $\sigma$-algebra $\mathscr{P}$ is the collection of sets generated by $\{\lambda \in \mathcal{P}(\Omega): \lambda(A) \in B\}$ where $A \in \mathscr{F}$ and $B$ is a Borel set of $\mathbf{R}$.

How can we prove that the map $x\mapsto \delta_x$ is measurable, where $\delta_x$ is the Dirac on $x$?

• I think you mean $\mathcal{P}(\Omega)$. (This notation is also a bit confusing since usually $\mathcal{P}$ is used to denote power set.)
– Ian
Commented Jan 4, 2016 at 14:31
• Thank you, it was a typo; about $\mathcal{P}$, do you have a better suggestion?
– user207096
Commented Jan 4, 2016 at 14:32
• You're right David; I added the definition of $\mathscr{P}$, which should coincide with the standard weak topology in the "standard" cases..
– user207096
Commented Jan 4, 2016 at 14:44
• "do you have a better suggestion?" A common notation for the set of probability measures on $\Omega$ is $$\mathcal M_1^+(\Omega).$$
– Did
Commented Jan 16, 2016 at 11:43

The set of $x$ such that $\delta_x$ lies in your generic measurable set is $\{x:\delta_x(A)\in B\}$, which in turn is one of the four sets $\emptyset$, $\Omega$, $A$ or $\Omega\setminus A$, depending on whether $0\in B$ and $1\in B$.