Summary table of probability terms

This page is a summary table of probability terms.

Table

Term	Symbol	Type	Definition
Reals	$R$
Borel subsets of the reals	$B$
Sample space	$Ω$
Outcome	$ω$	$Ω$
Events or measurable sets	$F$
Probability measure	$P$ or $Pr$ or $P_{F}$	$F \to [0, 1]$
Probability triple or probability space	$(Ω, F, P)$
Distribution	$μ$ or $D$ or $D$ or $P_{B}$ or $L (X)$ or $P X^{- 1}$	$B \to [0, 1]$	$B \mapsto P (X \in B)$
Induced probability space	$(R, B, μ)$
Cumulative distribution function or CDF	$F_{X}$	$R \to [0, 1]$
Probability density function or PDF	$f_{X}$	$R \to R$
Random variable	$X$	$Ω \to R$
Indicator of $A$	$1_{A}$	$Ω \to {0, 1}$
Expectation	$E$ or $E$	$(Ω \to R) \to R$

Let $(Ω, F, P)$ be a probability space.

Given a random variable, we can compute its distribution.
Given a random variable, we can compute the probability density function.
Given a random variable, we can compute the cumulative distribution function.
Given a distribution, we can retrieve a random variable. But this random variable is not unique? This is why we can say stuff like "let $X \sim D$ ".
Given a cumulative distribution function, we can compute the random variable. (Right?)
Given a probability density function, can we get everything else? Don't we just have to integrate to get the cdf, which gets us the random variable and the distribution?