Given any probability measure u on$(R,B(R))$, define an r.v. whose probability measure is u. Can this be done in an arbitrary probability space?

Question

I am a self-learner in this probability theory and measure theory area with very few analysis backgrounds. I am not so clear about this problem.

It seems that

$F(x)=P(X(\omega)<=x)=X(\omega)$, F is the distribution function and X is the random variable.

Can anyone give an explanation in detail? Or any other example. Many thanks!

Answer 1

The first equation $F(x)=\Pr(X\leq x)$ is correct. That is what it means to say that the distribution of the r.v. $X$ is $F$.

I'm not exactly sure what you are trying to say in the second equation. I think you are thinking of the measure-theoretic definition of the random variable X, so that $$\mu\left(\{\omega | X(\omega)\leq x\}\right)=F(x)$$

A random variable $X$ is a real number that somehow encodes the outcome of an experiment. We want to talk about the probability that $a<X<b$ for example. Measure theory is a way to put probability on a firm mathematical foundation, by defining a probability space as a measure space of total measure $1$, and a random variable as a measurable function on that space. I've always felt that, however technically effective this definition of a random variable is, it isn't very intuitive. This may be the reason that even though measure theory dates from the late $19$th and early $20$th centuries, it wasn't until $1933$ that Kolmogorov axiomatized probability theory in terms of measure theory.

This is just a personal opinion, and others may disagree.

EDIT

I didn't realize that the title was supposed to be the question. It's usually plainer if you put the whole question in the question body, but I should have paid more attention.

Now I think I know what is meant by $X(\omega)=\omega$. If we take $X$ to be the identity function, then $\Pr(X\in B)=\mu(B)$ for any measurable set $B$, where $\mu$ is the given probability measure, so the distribution of the identity function is the given measure. When they ask whether it can be done on an arbitrary probability space, I'm not sure what they are asking. Obviously, if a random variable must be real-valued, this can't work unless the probability space is some subset of the real numbers. If we extend the definition of a random variable to be any measurable function, so that it can take values in any probability space, then the identity function also works. See https://en.wikipedia.org/wiki/Random_variable#Measure-theoretic_definition