Area under the curve

Measure-theoretical definition

Integrating a function w.r.t. a measure. Given a Measure space, we can use the characteristic function of a set, to define "simple" functions defined as taking a certain value $c_i$ for $x$ that lies in any of a family of {elements $A_i$ of the sigma-algebra (which is a subset)}, which are disjoint. The integral of a simple function $g$ is just the area under the graph, defined as $\int g d\mu :=\sum_{i=1}^N c_i \mu(A_i)$. Can extend for integral of non-negative Measurable functions $f$, as the {Supremum of {the integral of a simple function} over simple functions which lower bound the function $f$}