Editing Conditional expectation (section)

==== Uniqueness ====

'''Example 1''': Consider the case where {{mvar|Y}} is the constant random variable that is always 1.
Then the mean squared error is minimized by any function of the form
:<math>
e_X(y) = \begin{cases}
\mu_X & \text{if } y = 1, \\
\text{any number} & \text{otherwise.}
\end{cases}
</math>

'''Example 2''': Consider the case where {{mvar|Y}} is the 2-dimensional random vector <math>(X, 2X)</math>. Then clearly
:<math>\operatorname{E}(X \mid Y) = X</math>
but in terms of functions it can be expressed as <math>e_X(y_1, y_2) = 3y_1-y_2</math> or <math>e'_X(y_1, y_2) = y_2 - y_1</math> or infinitely many other ways. In the context of [[linear regression]], this lack of uniqueness is called [[multicollinearity]].

Conditional expectation is unique up to a set of measure zero in <math>\mathbb{R}^n</math>. The measure used is the [[pushforward measure]] induced by {{mvar|Y}}.

In the first example, the pushforward measure is a [[Dirac distribution]] at 1. In the second it is concentrated on the "diagonal" <math>\{ y : y_2 = 2 y_1 \}</math>, so that any set not intersecting it has measure 0.