I'm having trouble understanding this notation

Question

This is possibly a really simple question. Suppose you had a probability density $Q$ defined on $S_n$, and $A$ is a subset of $S_n$. My understanding is that $Q(A)$ would be defined as

$$ \{x \in \mathbb{R} \mid \exists \pi \in S_n, Q(\pi)=x\}, $$ where $Q(\pi)$ is the probability that $Q$ is $\pi$.

However, in this article:

http://www.ams.org/samplings/feature-column/fcarc-shuffle

the author seems to use that notation differently (specifically, before halfway through the page, just underneath the boxed theorem in the "Strong Uniform Stopping Rules" section, when he talks about the proof of it and uses $Q^k(A)$). Can anyone explain?

Answer 1

You are confusing two different meanings for the same notation.

Your first is the function image notation: $Q:X\to Y$ and $A\subseteq X$ then $Q(A)=\{y\in Y: \exists x\in X, Q(x)=y\}$.

But that is not usually the meaning used when $Q$ is a probability density.

When $Q$ is a probability density, then $Q(A)$ usually means, with $Q(A)=\sum_{x\in A} Q(x)$.

Same notation, different meaning in different contexts. That happens sometimes in math.