What exactly does this physically mean?

Question

Let X(w) be a real random variable on ($\Omega$ , P). The image X($\Omega$) the set of all the values X(w) can take ,written $\Omega^{X}$. For any set $ B \subset \Omega^{X}$ the probability of the event that the value of X lies in B is equal to $P\{w|X(w) \in B\} = P(X^{-1}(B))$.

What I understood from that is that the probability measure P is the probability of the set of all ws such that the event X(w) is realized and which from that is an element of B, namely is equal to $P\{w|X(w) \in B\}$ however I do not understand why $P(X^{-1}(B))$ is used? Does it actually mean the inverse of event X or inverse composite? How physically can I vindicate the use of $X^{-1}$? Any ideas?

Answer 1

The sentence beginning "What I understood" is difficult to parse, but from the last part of it I get the impression that you are unaware that $\{w\mid X(w)\in B\}$ is the same thing as $X^{-1}(B)$. If that's the problem, then you should review the definition of $X^{-1}(B)$, perhaps not in a specifically probabilistic context but in the general context of any function $X$ that maps into a set that includes another set $B$ as a subset.

Answer 2

Several inconsistencies, which might hinder your understanding:

Let X(w) be a real random variable on ($\Omega$ , P).

No, a real random variable is a mapping $X:\Omega\to\mathbb R$, not the image $X(\omega)$ of a specific element $\omega$ of $\Omega$ by the mapping $X$.

The image X($\Omega$) the set of all the values X(w) can take ,written $\Omega^{X}$.

This is a very bad idea to use $\Omega^X$ to denote $X(\Omega)=\{X(\omega)\mid\omega\in\Omega\}$. Let us stick to $X(\Omega)$.

For any set $ B \subset \Omega^{X}$ the probability of the event that the value of X lies in B is equal to $P\{w|X(w) \in B\} = P(X^{-1}(B))$.

Yes, replacing $ B \subset \Omega^{X}$ by $B\subseteq\mathbb R$ (and in fact, by $B\in\mathcal B(\mathbb R)$ the Borel sigma-algebra on $\mathbb R$).

What I understood from that is that the probability measure P is the probability of the set of all ws such that the event X(w) is realized and which from that is an element of B, namely is equal to $P\{w|X(w) \in B\}$ however I do not understand why $P(X^{-1}(B))$ is used? Does it actually mean the inverse of event X or inverse composite? How physically can I vindicate the use of $X^{-1}$? Any ideas?

This is a quite general notation, not specific to probability theory: if $X:\Omega\to\mathbb R$ and $B\subseteq\mathbb R$, then $X^{-1}(B)=\{\omega\in\Omega\mid X(\omega)\in B\}$. Thus, such that the event X(w) is realized and inverse of event X mean nothing: $X$ is not an event (events are (some) subsets of $\Omega$) and $X^{-1}$ does not mean an inverse of $X$. Rather, for every $B\subseteq\mathbb R$, $X^{-1}(B)\subseteq\Omega$ is defined as above.