Interpretation of Independence of random variables

Question

A definition given in textbook:

Let $Y_i: \Omega \rightarrow \Omega_i$ be the ith projection onto $(\Omega_i, F_i)$. Family $(Y_i)_{i\in I}$ is Independent with respect to P if for an arbitrary choice of $B_i\in F_i$, the family of events $(\{Y_i \in B_i\})_{i\in I}$ is independent.

I am not very sure about the meaning of $\{Y_i\in B_i\}$. I have seen somewhere that it is just the event that $B_i$ occurs.

But how do I relate the interpretation to the random variable $Y_i$?

Answer 1

$\{Y_i\in B_i\}$ is an abbreviation of the set $\{\omega\in\Omega\mid Y_i(\omega)\in B_i\}$ which can be recognized as the preimage of set $B_i$ under function $Y_i:\Omega\to\Omega_i$.

Another notation for it is $Y_i^{-1}(B_i)$.

In your question you name the $Y_i$ as projections, so I suspect that we are dealing with the product $\Omega=\prod_{i\in I}\Omega_i$ here which can be looked at as the set of all functions $\omega:I\to\bigcup_{i\in I}\Omega_i$ that have the special property:$$\forall i\in I\;[\omega(i)\in\Omega_i]$$

Then $Y_i:\Omega\to\Omega_i$ is the function prescribed by $\omega\mapsto \omega(i)$ so that: $$\{Y_i\in B_i\}=\{\omega\in\Omega\mid \omega(i)\in B_i\}$$ Often $\omega(i)$ is abbreviated as $\omega_i$ leading to:$$\{Y_i\in B_i\}=\{\omega\in\Omega\mid \omega_i\in B_i\}$$

Answer 2

$(Y_i\in B_i)$ is simply the inverse image $Y_i^{-1}(B_i)$. So the definition says $Y_i$'s are independent iff $$P(Y_1^{-1}(B_1) \cap Y_2^{-1}(B_1) \cap ... \cap Y_n^{-1}(B_1))$$ $$=P(Y_1^{-1}(B_1))P(Y_2^{-1}(B_1))...P(Y_n^{-1}(B_1))$$ for all choices of Borel sets $B_1,B_2,..,B_n$.