izak

note: probability measure vs. random variable

The measure $P_X$ induced by a random variable $X$ (its distribution) describes the probabilities of $X$ taking values in measurable sets but does not fully specify the underlying probability measure $P$ on $(\Omega, \mathcal{F}, P)$. This is because $P_X$ only captures the behavior of $X$ via $P_X(A) = P(X^{-1}(A))$, ignoring other aspects of $\Omega$ and $P$. Different probability spaces (say discrete $\Omega_1 = \{0,1\}$ vs. continuous $\Omega_2 = [0,1]$) can yield the same $P_X$ (say Bernoulli(p)). And, $P_X$ loses information about dependencies between multiple random variables (say $X$ and $Y$ may have the same marginals but different joint distributions) and events unrelated to $X$. So, $P_X$ is a "pushforward measure" summarizing $X$’s distribution but omitting details about $\Omega$, non-$X$ events, and relationships with other variables. Only when $X$ completely determines $\Omega$ (say $X$ is the identity on $\mathbb{R}$) does $P_X$ fully characterize $P$.

More specifically, events not expressible in terms of a random variable $X$ are those that cannot be written as $\{X \in A\}$ for any measurable set $A$. These include events depending on other random variables (say $\{Y > 0.5\}$ when $X$ and $Y$ are defined on $[0,1]^2$), events relying on $\Omega$'s structure (say $\{\omega \text{ is even}\}$ when $X$ groups even and odd outcomes together), or events in product spaces where $X$ depends on only one component (say $\{\omega_2 > 0.5\}$ when $X = \omega_1$). Path-dependent events (say "Brownian motion hits 1 before $t=1$") or "meta" events (say "an infinite binary sequence contains infinitely many 1s") also cannot be described by $X$ alone. Since $P_X$ only encodes probabilities of $\{X \in A\}$-type events, it cannot capture these broader aspects of $(\Omega, \mathcal{F}, P)$. So, $P_X$ alone is insufficient to reconstruct the full probability measure $P$.