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Consider the following generalization of conditional expectation:

Start with measurable spaces $A,B$ equipped with measures $\mu_A, \mu_B$ respectively and a measurable map $T : A \to B$ s.t. $\mu_A \circ T^{-1} \leq C \mu_B$ for some $C > 0$ (i.e. this inequality holds event-wise).

Then $T$ induces (in a functorial, contravariant way) a linear map

$$\mathcal L_2(T) : \mathcal L_2(B,\mu_B) \to \mathcal L_2(A,\mu_A), g\mapsto g\circ T.$$

Indeed

$$\|g\circ T\|_2 = \int g(Tx)^2\,d\mu_A(x) \leq \int g(y)^2 \,dC\mu_B(y) = C\|g\|_2,$$

which shows that $\mathcal L_2(T)$ is well-defined and bounded with operator norm $\leq C$ (in fact, it is an isometry if $\mu_A \circ T^{-1} = \mu_B$).

Then $\mathcal L_2(T)$ has an adjoint operator $\mathbb E^T$.

If we consider a probability space $(\Omega, \mathcal F, \mathbb P)$ and sub-$\sigma$-algebra $\mathcal G$ of $\mathcal F$. and set $A = (\Omega,\mathcal F)$, $B = (\Omega, \mathcal G)$, $\mu_A = \mathbb P$, $\mu_B = \mathbb P_{|\mathcal G}$ and $T(t) = t$, the defining property of adjoints implies

$$\mathbb E(\mathbb E^T X \cdot \mathbb 1_B) = \mathbb E(X\cdot \mathbb 1_B)$$

for all $B\in \mathcal G$. So $\mathbb E^T$ is the conditional expectation of $X$ w.r.t. $\mathcal G$.

Are there any important examples of $\mathbb E^T$ besides conditional expectation?

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  • $\begingroup$ Do you claim that abstract nonsense and important examples are contradictory? $\endgroup$ – Jochen Jun 12 '18 at 8:19
  • $\begingroup$ @Jochen Ah no, point taken. $\endgroup$ – Stefan Perko Jun 12 '18 at 10:32
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Under mild conditions on the measurable spaces $A$ and $B$, the meausre $\mu_A$ admits a disintegration $(t,C)\mapsto K(t,C)$ (with $K(\cdot,C)$ a $B$-measurable function for each $A$-measurable set $C$, and $K(t,\cdot)$ a probability on $A$ for each $t\in B$) such that $$ \int_B f(t)K(t,C)\,d\nu(t) =\int_A f(T(a))1_C(a)d\mu_A(a), $$ where $\nu:=\mu_A\circ T^{-1}$. One can interpret $a\mapsto K(T(a),C)$ as the $\mu_A$-conditional probability of $C$ given $T^{-1}(\mathcal B)$, where $\mathcal B$ is the $\sigma$-field on $B$. [A good reference for this is the article "Conditioning as disintegration" [Statist. Neerlandica 51 (1997) 287–317] of Chang and Pollard.] In this sense the answer to your question is NO.

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