In what sense are usual probability distributions like normal, binomial, etc distributions in the formal definition?

Question

I just learned the formal definition of a distribution: a linear continuous functional $T$ from the vector space of test functions $C_{c}^{\infty}$ to its field of scalars $\mathbb{K}$. Ok, I guess.

But how do usual probability distributions fit this definition? For example, how is the normal distribution assigning a function in $C_{c}^{\infty}$to a real number in $\mathbb{K}$?

Bonus (what I'm really trying to understand when studying this): what does it formally mean for a random variable $X$ to have a distribution $T$? What is formally going on when I say something like $X \sim \mathcal{N}(\mu, \sigma^2)$?

Answer 1

The word "distributions" has two meanings, in two overlapping communities:

1. A probability measure. Specifically, Given a probability space $(\Omega,{\mathcal F}, P)$, the distribution of a random variable (i.e., a Borel measurable function) $X:\Omega \to {\mathbb R}$ is the probability measure $\mu$, defined on on Borel sets of ${\mathbb R}$ by $\mu(B)=P(\{\omega \in \Omega: X(\omega) \in B\})$. The right hand side is usually abbreviated as $P(X \in B)$ or $P\circ X^{-1}(B)$.

https://en.wikipedia.org/wiki/Probability_distribution

2. A linear functional on a suitable space of smooth functions, often with additional restrictions (e.g. "tempered distributions".)

Measures are distributions in the latter sense, but many distributions do not correspond to measures.

https://en.wikipedia.org/wiki/Distribution_(mathematics)