Equation (11.1) of these notes on concentration inequalities contains the equality $$\sum_{i=1}^n\Bbb E[Z-Z_i]^2_{+/-}=\sum_{i=1}^n\Bbb E\left[Z-\Bbb E^{\{i\}}[Z]\right]^2$$ but the $+/-$ subscript notation under the expectation is not explained.
Here $Z$ is a function of random variables $X_i,i\le n$, and $^{\{i\}}$ means swapping $X_i$ for an independent copy $X_i'$ but keeping all $X_j,j\ne i$ the same.
The equality implies that somehow, the notations $Z_i$ (which means a replacement of $Z$ by $X_i'$ at its $i$th value) and $+/-$ together mean a conditional expectation on $Z$. The proofs further down the page do not use this notation and instead prove the inequality for the other expression.
A subscript $+$ by itself can mean a positive domain only, and likewise for $-$, but I have never seen them used together before. I was suggested a possible "signed measure" interpretation; however it is usually denoted with a superscript and doesn't seem applicable in this context. Any clarification will be appreciated.
Best Answer
From (10.3) and (10.5) in the previous notes we have $$\frac12\Bbb E(Z - Z_i)^2 =\Bbb E(Z - Z_i)_-^2 =\Bbb E(Z - Z_i)_+^2 =\Bbb E\bigl(Z -\Bbb E^{\{i\}}Z_i\bigr)^2.$$ Comparing that with (11.1) suggests that $\Bbb E(Z - Z_i)_{+/-}^2$ means $\Bbb E(Z - Z_i)_+^2$ or $\Bbb E(Z - Z_i)_-^2$.