Projection interpretation of conditional expected value

Let $𝑋, 𝑌 : 𝜉 \to ℝ$ be real random variables. For any function $ℎ : ℝ \to ℝ$ , #m/thm/prob

𝔼 [(𝑌 - 𝔼 [𝑌 ∣ 𝑋]) ℎ (𝑋)] = 0

implying $C o r r [𝑌 - 𝔼 [𝑌 ∣ 𝑋], ℎ (𝑋)] = 0$ , i.e. $𝑌 - 𝔼 [𝑌 ∣ 𝑋]$ is completely uncorrelated with any function of $𝑋$ . This motivates the following interpretation of conditional expectation:

$𝔼 [𝑌 ∣ 𝑋]$ contains the entire component of $𝑌$ which is a function of $𝑋$ (i.e. determined by $𝑋$ ) and the residual component $𝑌 - 𝔼 [𝑌 ∣ 𝑋]$ is completely uncorrelated with any function of $𝑋$ .

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