Editing Conditional expectation (section)

==== Conditional probability ====

{{Main|Regular conditional probability}}

For a Borel subset {{mvar|B}} in <math>\mathcal{B}(\mathbb{R}^n)</math>, one can consider the collection of random variables
:<math> \kappa_\mathcal{H}(\omega, B) := \operatorname{E}(1_{X \in B}|\mathcal{H})(\omega). </math>
It can be shown that they form a [[Markov kernel]], that is, for almost all <math>\omega</math>,
<math>\kappa_\mathcal{H}(\omega, -)</math> is a probability measure.<ref>{{cite book |last1=Klenke |first1=Achim |title=Probability theory : a comprehensive course |date=30 August 2013 |location=London |isbn=978-1-4471-5361-0 |edition=Second}}</ref>

The [[Law of the unconscious statistician]] is then
:<math> \operatorname{E}[f(X)\mid\mathcal{H}] = \int f(x) \kappa_\mathcal{H}(-, \mathrm{d}x), </math>
This shows that conditional expectations are, like their unconditional counterparts, integrations,
against a conditional measure.