*Def Probability space/Sample space: A measure space $(\Omega,\mathcal{F},p)$ where $\Omega$, the sample set, is the set of all possible outcomes in a given experiment. $\mathcal{F}$ is a sigma algebra defined on the set $\Omega$ and $p$ is a measure on the measurable space $(\Omega,\mathcal{F})$ such that it countably additive and the measure of the whole sample set is $1$.
*Def Random variable: A random variable is a measurable function $X$ defined on $(\Omega,\mathcal{F},p)$ such that $X : \Omega \to \mathbb{R}$, the measure space $\mathbb{R}$ with Lebesgue measure?
*Def Push Forward measure: For a random variable it is defined by $p_{X}(A)=p(X^{-1}(A))$ for any measurable set $A$ of $\mathbb{R}$.
My question is that under these set of definition can I view Random variable with its push forward measure as something that 'Transforms' the given probability space into a probability space defined by the triplet $(\mathbb{R},\mathcal{F^{'}},p_X)$ where $\mathcal{F}'$ is the set of all measurable subsets of $\mathbb{R}$. Notice that $\mathbb{R}$ can be replaced with $Im(\Omega)$ under $X$ with suitable adjustments to the measure.
In short, have I just decided to view my old space under a new lens where I have reduced the amount of unrequired information? And when we talk about the transformation of a random variable, are we ultimately transforming the probability spaces into new ones with their natural push forward measures?
Best Answer
Yes, you could think of the pushforward measure as giving you a particular view of your original probability space. How much information it gives you depends on the random variable $X$ (e.g., $X=\text{const}$ gives you no information whatsoever).
A transformation of a random variable, say $f(X)$, is just a function of your random variable. So it gives you another function $\Omega \to \mathbb{R}$, i.e., another random variable. This random variable comes with its own pushforward measure, which as above can be thought of as giving you certain information about your original probability space.