A probability space consists of three components: A sample space, a $\sigma$-algebra, and a probability measure.
$$ \big(\Omega, \mathcal{F}, \mathbb{P}\big) $$
$$ \big(\Omega, \mathcal{F}, \mathbb{P}\big) \overset{X}{\longrightarrow} \big(\mathcal{R}, \mathcal{B}(\mathcal{R}), \mathbb{P} \circ X^{-1}\big) $$
Sometimes you come across a paragraph that you don’t fully understand, but you recognize that it is important and so your initial thought — how do I learn more to really appreciate this!
<aside>
“Measure theory for its own sake is based on the fundamental addition rule for measures. Probability theory supplements that with the multiplication rule which describes independence; and things are already looking up. But what really enriches and enlivens things is that we deal with lots of σ-algebras, not just the one σ-algebra which is the concern of measure theory.”
</aside>
Can you show that correlation at the population level implies correlation within the sample under i.i.d sampling?
$Y \perp X$ is equivalent to saying the following:
$$ \forall A \in \sigma(Y), B\in \sigma(X),\quad \mathbb{P}(A \cap B) = \mathbb{P}(A)\mathbb{P}(B) $$