Help me decide on notation that would resolve the following collision. Thanks.
Consider random variables $X,Y$ defined on a probability space $(\Omega,\Sigma,\mu)$. My question partly applies to general measures $\mu$, which is why I don't use $P$ here. When we define the conditional expectation via the Radon-Nikodym derivative, we deal with a restriction of a measure $\boxed{\mu_{|\mathcal{H}}}$. To be specific, consider a sub-sigma-algebra $\mathcal{H}\subseteq\Sigma$, e.g. the induced sigma-algebra $\mathcal{H}:=\sigma(Y)$, define $\nu(H):=\int_{H} X \mathrm{d}\mu$ and $\mu_{|\mathcal{H}}(H):=\mu(H)$ for $H\in\mathcal{H}$, then $\mathbb{E}[X|\mathcal{H}]:=\frac{\mathrm{d}\nu}{\mathrm{d}\mu_{|\mathcal{H}}}$. In this example we need the restriction because the Radon-Nikodym derivative technically requires both measures to be on the same measurable space, in this case $(\Omega,\mathcal{H})$. In many texts, the push-forward of $\mu$ by $X$ (the induced distribution) is denoted by $\mu_X\equiv X_*\mu\equiv \mu\circ X^{-1}$. It should be possible to come up with an example for a restriction of a pushforward $\boxed{\mu_{X|\mathcal{H}}}$.
A different and incompatible usage of the vertical bar arises in conditioning. We already saw $\mathbb{E}[X|\mathcal{H}]$, which does not cause a problem per se. A true collision takes place when we consider regular conditional distributions and probabilities [Wiki]. For a probability measure $\mu$, a r.v. $X$, and sub-sigma-algebra $\mathcal{H}$, I want to define a regular conditional distribution $\boxed{\mu_{X|\mathcal{H}}}$. Further, in some cases I may be lucky to have a nice measurable space, so that with $X=\mathrm{id}$ I obtain a regular conditional probability $\boxed{\mu_{|\mathcal{H}}}$.
Here are some options I thought of, but did not find very satisfactory:
- If I use $\mu_{X,\mathcal{H}}$ and $\mu_{\mathcal{H}}$ for r.c.d. and r.c.p., then when conditioning on a random variable, I would be tempted to write $\mu_Y$, which could be interpreted ambiguously as either the r.c.p. $\mu_{\sigma(Y)}$ or the push-forward $Y_*\mu$.
- Some authors use $\mu_X(\cdot|\mathcal{H})$ and $\mu(\cdot|\mathcal{H})$ for a r.c.d. and r.c.p. One disadvantage is the extra symbols, another is that I already use the same notation for the conditional probability $\mu(E|\mathcal{H}):=\mathbb{E}[1_E|\mathcal{H}]$, which is not necessarily regular. Authors who use this notation have to make it clear if they imply the regular version of the conditional probability or distribution, which in principle may not exist.
- Billingsley, for example, uses the double bar for conditioning, as in $\mathbb{E}[X\|\mathcal{H}]$, but double bars are not as handy and popular as single bars.
- Some authors, including Terrence Tao, use $\downharpoonright$, $\restriction$ or $|$ for the restriction of sets and functions, in particularly, measures. Theses symbols are not subscripts. The downside is that $\mu|_{\mathcal{H}}$ and $\mu_{|\mathcal{H}}$ look similar in handwriting. Also, I can't come up with a mnemonic to remember which one is restriction or conditioning.
Bonus question. Should I use $B_{|x_1}$ for a cross-section of $B\subseteq \mathbb{R}^2$ at $X_1=x_1$? That is, $B_{|x_1}:=\{x_2:(x_1,x_2)\in B\}\subseteq \mathbb{R}$. How do I denote the projection on one axis then? In my opinion, cross-section to projection is like conditioning to restriction.