Definition:Random Variable/Real-Valued/Definition 2

Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\map \BB \R$ be the Borel $\sigma$-algebra on $\R$.

Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {\R, \map \BB \R}$.

Then we say that $X$ is a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.