How to use advanced probability theory to explain "Independent" of two randan variable?

Question

We know from elementary probability theory, two random variable $X,Y$ are called independent if $F_X(x)F_Y(y)=F_{XY}(x,y).$ where the notation are their cdf. But in measure space, assume $X,Y$ $\sim$ $(\Omega,\mathcal{F},\mathcal{P}).$ If we take $\Omega=[0,1],$ suppose $w_1\in \Omega=[0,1]$ and $X(w_1)=1$, how to explain $Y$ in this $(\Omega,\mathcal{F},\mathcal{P})$ in order to make $X$ and $Y$ independent? Can you give me an example or some related theorems to help me deeply understand this? Thank you so much!

Answer 1

Here is an answer from a general measure-theoretic point of view.

If you have real-valued random variables $X,Y$ on $\Omega$, you can use them to obtain two (probability) measures on ${\bf R}$, namely $X[\mathcal P]$ and $Y[\mathcal P]$. At the same time, $(X,Y)$ is a ${\bf R}^2$-valued random variable on $\Omega$, which gives us a (probability) measure $(X,Y)[\mathcal P]$ on ${\bf R}^2$.

We say that $X,Y$ are independent if $(X,Y)[\mathcal P]=X[\mathcal P]\otimes Y[\mathcal P]$, or in other words, $(X,Y)[\mathcal P]$ is the product measure of $X[\mathcal P]$ and $Y[\mathcal P]$. By definition, it means that for any intervals $I,J\subseteq {\bf R}$ you have $$ \mathcal P((X,Y)^{-1}[I\times J])=\mathcal P(X^{-1}[I])\cdot \mathcal P(Y^{-1}[J]) $$ Since $(X,Y)^{-1}[I\times J]=X^{-1}[I]\cap Y^{-1}[J]$, this is equivalent to saying that for any intervals $I,J$ you have $$ \mathcal P(X\in I\land Y\in J)= \mathcal P(X\in I)\cdot \mathcal P(Y\in J). $$ I.e. the events $X\in I$ and $Y\in J$ are independent. It is easy to see that you can take half-lines instead of intervals here, so in fact this is equivalent to saying that for any real numbers $a,b$ the events $X\leq a$ and $Y\leq b$ are independent, which is exactly the statement you have given.

The first definition generalises immediately to random variables taking value in an arbitrary measure space (except then you may have no notion of a cumulative distribution function, or you may have to consider more complicated sets than just intervals), and in fact measurable functions on any measure space. It also works more or less the same for more than two variables (even infinitely many).