In measure theory, a simple function is a function that is a finite linear combination $$ h = \sum_{k=1}^n c_k \chi_{A_k} $$ of characteristic functions, where the $c_k$ are real coefficients and every $A_k$ is a measurable set with respect to a fixed measure space.

If the measure space is $\mathbb{R}$ and each $A_k$ is an interval, then the function is called a step function. Thus, every step function is a simple function.

Simple functions are used in analysis to interpolate between characteristic functions and measurable functions. In other words, characteristic functions are easy to integrate: $$ \int_E \chi_{A}\,dx = |A|, $$ while simple functions are not much harder to integrate: $$ \int_E \sum_{k=1}^n c_k \chi_{A_k}\,dx = \sum_{k=1}^n c_k |A_k|. $$ To integrate a measurable function, one approximates it from below by simple functions. Thus, simple functions can be used to define the Lebesgue integral over a subset of the measure space.