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Let $(\Omega, {\cal B}, P )$ be a probability space. Let $( \mathbb{R}, {\cal R})$ be the usual measurable space of reals and its Borel $\sigma$- algebra. Let $X: \Omega \rightarrow \mathbb{R}$ be a random variable.

I am wondering if the following is correct: the probability distribution $Q$ of the random variable $X$ is a measure on $( \mathbb{R}, {\cal R})$ such that $Q( A ) = P( \{ w: X(w) \in A \} )$ for $A \in {\cal R}$.

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Yes, the distribution of a random variable $X:\Omega\to\mathbb{R}$ is indeed a probability measure (or probability distribution) on $(\mathbb{R},\mathcal{R})$ and is often called the pushforward measure of $P$ by $X$ and denoted by $P\circ X^{-1}$.

Note that it is well-defined, since $X$ is $(\mathcal{B},\mathcal{R})$-measurable and so $X^{-1}(A)\in\mathcal{B}$ for all $A\in\mathcal{R}$. Hence it makes sense to assign the probability of the sets $X^{-1}(A)$ for all $A\in\mathcal{R}$, and $$ P_X(A):=P(X^{-1}(A)),\quad A\in\mathcal{R}, $$ defines a probability measure on $(\mathbb{R},\mathcal{R})$. This follows from basic properties of the pre-image and from the fact that $P$ is a probability measure (try showing this yourself).

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