# Precise Definition of $\mathbb{E}[X\mid \sigma(A)]$ Conditional Expectation of Random Variable given Sigma Algebra generated by a set

I want to define precisely, exhaustively and constructively the conditional expectation of a random variable given the sigma algebra generated by a set. This question has some discussion on it but the definition is not made explicit.

### Set Up

Let $$(\Omega, \mathcal{F}, \mathbb{P})$$ be a probability space, $$(\mathsf{E}, \mathcal{E})$$ be a measurable space (Banach space should be enough) and $$X:\Omega\to\mathsf{E}$$ be a random variable whose expectation exists and $$\mathbb{E}[|X|] < \infty$$. Let $$\mathsf{A}\in\mathcal{F}$$ be an event and define the sigma algebra generated by it as $$\sigma(\mathsf{A}) := \{\emptyset, \Omega, \mathsf{A}, \mathsf{A}^c\}.$$ The conditional expectation $$\mathbb{E}[X\mid \sigma(\mathsf{A})]:\Omega\to\mathsf{E}$$ is a $$\sigma(\mathsf{A})$$-measurable random variable such that $$\int_\mathsf{B} X(\omega) \mathbb{P}(d\omega) = \int_\mathsf{B}\mathbb{E}[X\mid \sigma(A)](\omega) \mathbb{P}(d\omega) \qquad\qquad\forall\, \mathsf{B}\in\{\emptyset, \Omega, \mathsf{A}, \mathsf{A}^c\}. \qquad\qquad (*)$$ I would like to construct this random variable. For instance, when $$\mathsf{A} = \Omega$$ and $$\sigma(\mathsf{A}) = \{\emptyset, \Omega\}$$ then it's easy to show that the constant random variable $$Y(\omega) = \mathbb{E}[X]$$ does the job. When $$\mathsf{A}\neq \Omega$$ things get a bit more complicated. We aim means that to find a version of $$\mathbb{E}[X\mid \sigma(\mathsf{A})]$$, that is a random variable $$Y:\Omega\to\mathsf{E}$$ that is $$\sigma(\mathsf{A})$$-measurable and satisfies $$(*)$$. I cannot show that either holds (constructively). Even simply showing measurability is tricky because I don't know how to construct this random variable.

##### Measurability

Most measure theory books define the conditional expectation $$\mathbb{E}[X\mid \mathsf{A}]$$ to be a number, typically $$\mathbb{E}[X \mid \mathsf{A}] = \begin{cases} \displaystyle \frac{1}{\mathbb{P}(\mathsf{A})}\int_\mathsf{A} X(\omega) \mathbb{P}(d\omega) & \text{ if } \mathbb{P}(\mathsf{A}) > 0 \\ \text{any value in } \mathsf{E} & \text{ if } \mathbb{P}(\mathsf{A}) = 0. \end{cases}$$ My guess is that this number must appear somewhere in our definition of the random variable $$Y(\omega) = \mathbb{E}[X\mid \sigma(\mathsf{A})](\omega)$$. Recall that for $$Y(\omega)$$ to be $$\sigma(\mathsf{A})$$-measurable we require that for any $$\mathsf{C}\in\mathcal{E}$$ $$Y^{-1}(\mathsf{C}) \in \{\emptyset, \Omega, \mathsf{A}, \mathsf{A}^c\}.$$ Now when $$\mathbb{P}(\mathsf{A}) > 0$$ then one has that $$Y^{-1}(\mathsf{C})\in\sigma(\mathsf{A})$$ because $$Y^{-1}(\mathsf{C}) = \{\omega\in\Omega\,:\, \mathbb{E}[X\mid \mathsf{A}]\in\mathsf{C}\} = \begin{cases} \emptyset & \text{if } \mathbb{P}(\mathsf{A})^{-1}\mathbb{E}[X 1_\mathsf{A}]\in\mathsf{C}\\ \Omega & \text{if } \mathbb{P}(\mathsf{A})^{-1}\mathbb{E}[X 1_\mathsf{A}]\notin\mathsf{C}. \end{cases}$$ When $$\mathbb{P}(\mathsf{A}) = 0$$ things get more tricky. Ideally I would want that for any $$\mathsf{C}\in\mathcal{E}$$ we have that $$Y^{-1}(\mathsf{C})$$ is either $$\mathsf{A}$$ or $$\mathsf{A}^c$$. I cannot seem to find a way of defining $$Y$$ so that this is true.

• For definitions see stats.stackexchange.com/a/450381/919. For examples of constructing a conditional expectation see stats.stackexchange.com/a/74339/919. For much more about this subject please search our site for the keywords: stats.stackexchange.com/…. A quick inspection of those hits suggests everything you ask here has answers already.
– whuber
Jun 1, 2022 at 13:23
• @whuber They define $\mathbb{E}[X\mid \mathcal{G}]$ for a general sub-sigma algebra $\mathcal{G}\subseteq \mathcal{F}$, but the point of my question is when that sigma algebra is the smallest sigma algebra containing a set $\mathsf{A}\subset\Omega$. I understand the general definition of conditional expectation! Jun 1, 2022 at 13:25
• That's a basic mathematical concept: that's what it means for one mathematical structure to generate another. Neither I nor, I suspect, most visitors to this page want to wade through the entire post to ferret out what you're trying to ask. Could you be short, explicit, and clear?
– whuber
Jun 1, 2022 at 13:26
• @whuber My question is simple: how do you construct the random variable $\mathbb{E}[X \mid \sigma(\mathsf{A})]$? I know it's just a special case of a more general definition, but I want to construct it Jun 1, 2022 at 13:29
• When $A$ is an atom, yes: the very definition of measurability (easily) implies any random variable must be constant on $A.$ That does not imply $X$ is a constant function on $\Omega$!
– whuber
Jun 2, 2022 at 13:40