Product-coordinate projections commute and contract in L2

Let $f\in L^2(\mu )$. Tonelli applied to $\mathrm{\mid }f{\mathrm{\mid }}^2$ shows that for almost every $x_J$, the section $z\mapsto f(x_J,z)$ is square-integrable on the complementary probability space. Cauchy–Schwarz with the constant function $1$ makes that section integrable, so [F1] defines its average there. If representatives $f$ and $\tilde{f}$ agree almost everywhere, Tonelli applied to $\mathrm{\mid }f-\tilde{f}{\mathrm{\mid }}^2$ shows their sections agree almost everywhere for almost every $x_J$; their averages therefore define the same almost-sure class. Fubini gives measurability of the section-average function.

On almost every section, Cauchy–Schwarz gives $\mathrm{\mid }{P}_{J}f(x_J){\mathrm{\mid }}^2\le \int \mathrm{\mid }f(x_J,z){\mathrm{\mid }}^{2}d{\mu }_{-J}(z)$ because ${\mu }_{-J}$ has mass one. Integrating in $x_J$ and using Tonelli yields $\mathrm{\parallel }{P}_{J}f{\mathrm{\parallel }}_2^2\le \mathrm{\parallel }f{\mathrm{\parallel }}_2^2$, so $P_J$ is a contraction. Averaging a function already independent of $z_{-J}$ leaves it unchanged, hence $P_J^2=P_J$.

For $f,g\in L^2$, Cauchy–Schwarz makes $fg$ integrable. Fubini in the two groups of coordinates gives $⟨P_{J}f,g⟩={\int }_{E_J}(P_{J}f)(P_{J}g)d{\mu }_J=⟨f,P_{J}g⟩$. Thus $P_J$ is self-adjoint.

Applying $P_K$ and then $P_J$ integrates over the coordinates outside $K$, then over the coordinates in $K\setminus J$. Repeated finite-product Fubini permits this order and identifies the iterated integral with integration over the complement of $J\cap K$. The equality holds almost everywhere by [F3], including when either set is empty or full, so $P_J{P}_{K}f=P_{J\cap K}f$ in $L^2$.