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!
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“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.”
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$Y \perp X$ is equivalent to saying the following:
$$ \forall A \in \sigma(Y), B\in \sigma(X),\quad \mathbb{P}(A \cap B) = \mathbb{P}(A)\mathbb{P}(B) $$