# Intuition about conditional probability given a $\sigma$-algebra [duplicate]

I've been studying some statistics by the ways of measure theory, and came up with a problem in understanding conditional probability. The book gives the following definitions:

Let $$(\Omega,\mathcal{F},\mathbb{P})$$ be a probability space, $$\mathcal{G} \subset \mathcal{F}$$ be a $$\sigma$$-algebra and $$A \in \mathcal{F}$$. Then, we define a probability measure on $$\mathcal{G}$$ by the law $$\nu(G) = \mathbb{P}(A \cap G).$$ If $$\mathbb{P}(G) = 0$$, then $$\nu(G) = \mathbb{P}(G \cap A) \leq \mathbb{P}(G) = 0,$$ hence $$\nu \ll \mathbb{P}$$. We define $$\mathbb{P}(A \mid \mathcal{G})$$ as $$d\nu/d\mathbb{P}$$. This way, the conditional probability is kind of well-defined, since Radon-Nikodym derivatives are equal almost everywhere.

My problem here is: I don't understand what is the intuition behind this definition. For some help in understanding my problem, there is an important result

The function $$\mathbb{P}(A \mid \mathcal{G})$$ has the following properties almost everywhere:

• $$0 \leq \mathbb{P}(A \mid \mathcal{G}) \leq 1$$;
• $$\mathbb{P}(\emptyset \mid \mathcal{G}) = 0$$;
• If $$\{A_n\}_{n = 1}^\infty$$ is a sequence of two-by-two disjoint sets in $$\mathcal{F}$$, then $$\mathbb{P}\left(\bigcup_{n=1}^\infty A_n \mid \mathcal{G}\right) = \sum_{n = 1}^\infty \mathbb{P}(A_n \mid \mathcal{G}).$$

This lemma says that, for almost every point, $$\mathbb{P}(A \mid \mathcal{G})$$ is a probability measure in $$\mathcal{F}$$ (as a function of $$A$$). Now, I don't have a single clue about what a "field" of probability measures mean and how it is related in any way with concepts of elementary probability. Even more, is there any "real-world" applications of this concept? With real-world applications I mean like some kind of elementary statistics problem that can be solved using this, some applicable situation that might appear in a non-math context.

• Could you please explain what "two-by-two disjoint" means and how it differs from "disjoint"? And what does it mean for a set function to be a probability measure at "almost every point"? For intuition on conditional expectations, please search our site.
– whuber
Dec 27, 2022 at 14:28