Product-coordinate averaging operators

Let $d\ge 1$ be finite, let $(E_j,{\mathcal{E}}_j,{\mu }_j)$ be probability spaces, and let $(E,\mathcal{E},\mu )={\prod }_{j=1}^d(E_j,{\mathcal{E}}_j,{\mu }_j)$ be their finite product with product sigma-algebra and product probability. Write $[d]=\{1,\dots ,d\}$. For $J\subseteq [d]$, let $E_J$ and ${\mu }_J$ denote the corresponding coordinate product; the empty product is a one-point probability space. For real $f\in L^2(\mu )$, define the coordinate average $P_{J}f(x_J)={\int }_{E_{[d]\setminus J}}f(x_J,z_{[d]\setminus J})d{\mu }_{[d]\setminus J}(z_{[d]\setminus J})$ for almost every $x_J$ for which the section is integrable, and assign zero on the exceptional null set. Regard $P_{J}f$ also as a function on the full product that is constant in the coordinates outside $J$. The conventions are $P_{[d]}f=f$ as an $L^2$ class and $P_{\mathrm{\varnothing }}f=\int fd\mu$.

The displayed integral is initially defined on representatives. Its independence of the representative and its $L^2$ properties are established by Product-coordinate projections commute and contract in L2 ↗.