“Probability measure” as a synonym of “Cumulative Distribution Function” (CDF)

probabilityprobability distributionsprobability theory

If

  1. a $\sigma$-algebra is a collection of subsets of a given set $\Omega$, and
  2. a "probability measure" is a function that assigns to each element of a given $\sigma$-algebra (i.e. to each subset of $\Omega$) a value between 0 and 1,

Can I use the term "probability measure" as synonym of "cumulative distribution function" (CDF)?

If yes, would it be correct to write, in a paper, something like this: "We call the two probability measures $\mu$ and $\nu$, as cumulative distribution functions $F$ and $G$, respectively?"

Or, am I saying a huge bull***t ?

Best Answer

Firstly, the CDF is not a measure, it is instead a monotone function.

Now, there is a 1-1 correspondence between probability measures on $\mathbb R$ and CDF's. That is, if $F$ is a CDF, then you can define a unique probability measure $\nu_F$ on $\mathbb R$ satisfying $\nu_F((-\infty,x))=F(x)$ for each $x$, and similarly, given the probability measure on $\mathbb R$ you define $F$.

However, if $X$ is a random variable on a probability space $(\Omega,\mu)$, with CDF $F$, then the aforementioned measure $\nu_F$ is NOT equal to $\mu$. It is instead the so-called "pushforward" of $\mu$ by the map $X$ (recall that random variables are just maps from $\Omega$ to $\mathbb R$ -- I am for simplicity only discussing real valued random variables here).

More precisely, $\nu_F(A)=\mu(X^{-1}(A))$ for every measurable $A\subseteq \mathbb R$.

So, if one were writing a paper, one might say that $F$ is the CDF associated with $X$, but that and $\mu$ are two totally different animals and should not be identified.

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