the distribution as a pushforward measure
When you say a random variable follows a Normal distribution, you are not talking about the messy underlying sample space at all — you are talking about how the variable spreads its weight across the real line. The pushforward measure is the rigorous form of this: it takes the probability living on the abstract space Omega and transports it through the random variable onto the real line, producing a probability measure on R that captures everything you can observe about the variable. This transported measure is called the distribution, or the law, of the random variable.
The construction is direct. Given a probability measure P on Omega and a random variable X: Omega -> R, define a new measure P_X on the Borel sets of R by P_X(B) = P(X^(-1)(B)) = P(X in B). In words: the weight P_X assigns to a region B on the line is simply the original probability of all the outcomes that X maps into B. You push the probability forward along X. Because X is measurable, X^(-1)(B) is always an event, so P_X(B) is always defined, and P_X is itself a genuine probability measure on (R, Borel). It is denoted X_* P or the law of X, and it lives entirely on the real line — the original Omega has been forgotten.
This is the precise reason the underlying probability space can be ignored in practice: two random variables with the same pushforward have identical distributions and behave identically for every probability statement, even if they were built on completely different spaces. Expectations, the CDF, densities, and moments are all computed from P_X alone. It also explains why we say things like let X be Normal(0,1) without ever describing Omega — we are simply naming the pushforward measure directly.
Roll a fair die on Omega = {1,...,6} with P uniform, and let X = 1 if the roll is even, else 0. The pushforward P_X lives on R: P_X({1}) = P({2,4,6}) = 1/2 and P_X({0}) = 1/2. The distribution of X is Bernoulli(1/2) — and we never again need to mention the die.
Probability transported from Omega onto the real line through X — the law of X, the only thing observation can see.
Equal distributions do not mean equal random variables: X and -X can share the same Normal(0,1) law (same pushforward) while never taking the same value on the same outcome.