2.2 Conditioning on sets of positive measure

This is probably the easiest and most intuitive notion of conditionals. Intuitively, this construction allows us to condition on events (or sets). Suppose \((\Omega, \mathcal{A}, \mathbb{P})\) is a probability space, and let \(B\in\mathcal{A}\) be a set of positive measure (i.e., \(\mathbb{P}(B) > 0\)). In that case, we can define a new probability measure \(\mathbb{P}(\cdot | B)\) on \(\mathcal{A}\) by the following (high school conditional measure):

\[ \begin{aligned} \mathbb{P}(A | B) := \frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)} \end{aligned} \]

It is straightforward to verify that this is a probability measure on \(\mathcal{A}\) . Note that, this notion of the conditional probability gives us a measure, and not a random variable, unlike some other notions of conditional expectations that we’ll see.