Simple function

A simple function is a function which takes finite numbers of values. Usualy in measure theory we mean the following:

Let $(\Omega ,\Sigma ,\mu )$ be a measure space. Let ${A_{1},...,A_{n}}\in \Sigma$ be a sequence of measurable sets, and let ${a_{1},...,a_{n}}$ be a sequence of complex numbers. A simple function is a function of the form $\sum _{k=1}^{n}a_{k}1_{A_{k}}$. Its Lebesgue integral is $\sum _{k=1}^{n}a_{k}\mu (A_{k})$.