Conditional expectation as a measurable function of the conditioning variable

The product function $(y,x)\mapsto f(x)$ is measurable by the rectangle inverse-image test. Thus [F3] makes h measurable. The rectangle identities of [F1] make $L(\omega ,A)=K(Y(\omega ),A)$ an RCD of X given $\sigma (Y)$, since the events of that sigma-algebra are exactly inverse images of measurable Y-events. Applying [F2] to this L gives $\int f(x)L(\omega ,dx)=E[f(X)\mid \sigma (Y)]$ as classes, under [F4]. The integral on the left is h(Y) by its definition, which proves the nonnegative clause, allowing infinity.

For real f, [F3] makes $a(y)=\int \mathrm{\mid }f(x)\mathrm{\mid }K(y,dx)$ and $D=\{a<\mathrm{\infty }\}$ measurable and makes the zero-filled signed h real measurable. The nonnegative test identity of [F1] gives $\int ad{P}_Y=E\mathrm{\mid }f(X)\mathrm{\mid }=C<\mathrm{\infty }$. For every integer $n\ge 1$, $n{P}_Y(D^c)\le C$, hence $P_Y(D^c)=0$. Its inverse image under Y is null by the marginal definition. On that inverse-image complement the signed integral through L is h(Y), and on it both zero-fill conventions agree. The signed clause of [F2] therefore proves the stated real integrable conditional identity. Values at any fixed marginal-null fibre are not prescribed by this almost-sure identity.