Definition:Random Variable/Continuous/Absolutely Continuous/Definition 1
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Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $P_X$ be the probability distribution of $X$.
Let $\map \BB \R$ be the Borel $\sigma$-algebra of $\R$.
Let $\lambda$ be the Lebesgue measure on $\struct {\R, \map \BB \R}$.
We say that $X$ is an absolutely continuous random variable if and only if:
- $P_X$ is absolutely continuous with respect to $\lambda$.
Sources
- 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $10$: Probability: $10.1$: Basics