Trying to get through https://terrytao.files.wordpress.com/2012/12/gsm-126-tao5-measure-book.pdf and I am already having trouble in the opening notation pages!
Page xi: "Given a subset $E$ of a space $X$, the indicator function $1_E : X \rightarrow R$ is defined by setting $1_E(x)$ equal to $1$ for $x \in E$ and equal to $0$ for $x \notin E$."
Can anyone explain what this is saying? I don't know what a "space" is, or what it means to have a subset of that space. Is a space just "some framework in which we define stuff" (not necessarily in the physical sense of space) and $E$ is "some collection of items defined within this space"?
I don't know why a function is named with a number. Am I understanding this right? Why isn't it $I_E$ or something?
Is this basically saying "Consider some $x$. Then $1_E(x) = 1$ if $x$ is in subset $E$ which we took from space $X$, and $0$ otherwise"? Is $1_E$ like a "role-call" function? You here $x$? No? $0$ for you.
Even if so, I still don't understand what $1_E : X \rightarrow R$ means.
In general, if you have two sets $X$ and $Y$, writing $f:X\to Y$ means that $f$ is a function from $X$ to $Y$. A space is a common term used to refer to a set with some additional structure. In this context, this structure would likely be a $\sigma$-algebra and a measure.
An indicator function $1_E$ is just a function that tells you whether a given point $x\in X$ is an element of $E$. In other words, $1_E(x)=1$ if $x\in E$, and $1_E(x)=0$ otherwise.
So for instance if $X=\mathbb{R}$ and $E=\mathbb{Z}$, we have $1_E(2)=1$ but $1_E(\frac{1}{2})=0$.