To clearly define integration, we will first define a few terms.
Definition 4.1.1 (Indicator function). Let $A\subset \R^n$be a bounded subset. The indicator function $1_A$ is
$$ 1_A(x) = \begin{cases}1 &x\in A\\ 0 & x \notin A . \end{cases} $$
This is used to integrate over subsets. For example, the expression $f(x)1_A(x)$ evaluates to 0 when $x\notin A$.
Definition 4.1.2 (Support of a function: Supp(f)). The support supp(f) of a function $f: \R^n \to \R$ is the closure of the set
$$ \{x\in \R^n\mid f(x)\neq 0\}. $$
Definition 4.1.3 ($M_A(f)$ and $m_A(f)$). If $A \subset \R^n$ is an arbitrary subset, we will denote $M_A(f)$ the supremum of $f(x)$ for $x\in A$, and $m_A(a)$ the infimum of $f(x)$ for $x\in A$. ****
Definition 4.1.4 (Oscillation). The oscillation of $f$ over $A$, denoted $\rm{osc}_A(f)$, is the difference between the supremum and infimum.
In 1 dimension, we learned that an integral is defined by cutting slices underneath a function to calculate its area. In n-dimensions, we will call these dyadic cubes. The idea of an integral is that it's a density function.
Definition 4.1.5 (Dyadic cube). A dyadic cube $C_{k,N} \subset \R^n$ is defined by the set
$$ C_{\bf k,N} = \{x\in \R^n\mid \frac{k_i}{2^N}\leq x_i<\frac{k_i+1}{2^N}\text{ for } 1\leq i \leq n\} $$
The first index, $\bf k$, locates each cube's lower left hand corner. $1/2^N$ tells the side length of each cube. As $N$ increases, the cubes become smaller and smaller.
Definition 4.1.7 (Dyadic paving). The collection of cubes $C_{k, N}$ at a single level $N$, denoted $\mathcal{D}_N(\R^n)$, is the $N$th dyadic paving of $\R^n$. It is a set of "half open" cubes.
$$ \mathcal{D}0(\R) = \{[k,k+1])\mid k\in \Z\}\\ \mathcal{D}1(\R) = \{[k,k+1/2])\mid k\in \Z\}\cup \{[k+1/2,k+1)\mid k \in \Z\}\\ = \{[1/2k,1/2k+1/2)\mid k \in \Z\}.\\ \mathcal{D}_0(\R^2)=\{[k,k+1)\times [l, l+1)\mid (k,l)\in \Z^2\}. $$