The Simple Functions and Its Properties.

Simple function

Let $(X,\sum)$ be a measurable space, A simple function on X is a function of the form
$\phi=\sum^n_{j=1}c_{j}X_Ej$, where for each $j-1,2,...,n$ and
$c_j$ is an extended real number and $E_j\epsilon\sum$.
A simple function is also a real valued function over a subset of the real line.
A simple function attains only a finite number of values.

Properties of the simple function

1. the sum of two simple functions is a simple function.
2. he difference and product of two simple functions is again a simple function.
3. multiplication by constant or scaler multiplication keeps a simple function simple.

Example

1. if $f:(X,\sum)\rightarrow\mathbb{R}^+$ is measurable then$sinf,exp(f)$ and $logf$ are also measurable on the set X on which they are definedand thus they are simple function.
2. if $f:(X,\sum)\rightarrow\mathbb{R}^+$ is measurable and $g:\mathbb{R}\rightarrow\mathbb{R}$ a continous function whose domain  contains the values f then the composition function $gof$ is measurable and thus simple.