Can someone explain the indicator function to me? 
Definintion. Let $(\Omega, \mathcal{A}, \mu)$ be a measure space and let $f : (\Omega, \mathcal{A}) \rightarrow (\mathbb{R}, \mathcal{B})$ be a nonnegative simple function with values $\alpha_1,...,\alpha_r \in \mathbb{R}.$ Define
  \begin{equation}
A_k = f^{-1}(\alpha_k),  k = 1,...,r,
\end{equation}
  so that the sets $A_k$ are measurable, pairwise disjoint, and the function $f$ can be written as
  \begin{equation}
f = \displaystyle \sum^r_{k = 1} \alpha_k \mathbf{1}_{A_k}
\end{equation}
  The integral $\int_\Omega f d\mu$ of $f$ over $\Omega$ with respect to $\mu$, is defined by
  \begin{equation}
\int_\Omega f d\mu = \displaystyle \sum^r_{k = 1} \alpha_k \mu(A_k)
\end{equation}

This is the first time the $\mathbf{1}$ is used, and I don't quite get what it means. There isn't any further explaining about the notation, but it's used very often in the homework assignments. Can someone explain to me why this notation is used or perhaps some literature where this is better explained?

Given a set $X$, the indicator function on $X$ is denoted $1_X$ or $1[X]$, and it is the function into $\{0,1\}$ given by $1_X(x) = 1$ if $x \in X$, and $1_X(x) = 0$ otherwise. The $1$ is meant to denote "this function takes the value $1$ if its argument is in…". The domain of the function is determined by context; in the instance you gave, the domain is $\mathcal{A}$.
It can be alternatively viewed as a way of picking out elements of a set. Indeed, for every subset $A \subseteq X$, there is an indicator function $1_A : X \to \{0,1\}$, and $A = 1_A^{-1}(\{1\})$.
