I am having a problem in understanding clearly what simple function actually means .
Royden says:
A real-valued function $\phi$ is called simple if it is measurable and
assumes only a finite number of values. If $\phi$ is simple and has the
${\alpha_1,\alpha_2,…..\alpha_n}$
values then $\phi=\sum_{i=1} ^n \alpha_i\chi_{A_i}$, where $A_i=${x:$\phi$(x)=$\alpha_i$}.
first question is : does $\phi$ have measure zero ? ( because it has a finite number of elements)
Why do we write the simple function in such a linear combination ?
Suppose if i have to write a general function in terms of a simple function, what should i take care of , so that a function can be written in terms of simple function?
Is there a geometric presentation to understand this concept ?
Thanks for the help!
Best Answer
Since $\phi$ is a function (and not a subset of the measure space) we can't really speak of its measure.
Simple functions are sort of "step functions" in the following sense; the sets $\{A_{i}\}_{i=1}^{n}$ form a partition of the measure space $X$ and $\phi$ takes a constant value in each $A_{i}$, i.e. $\phi|_{A_{i}}=\alpha_{i}$ for all $i$. The most economical way of writing this is through the indicator functions of each $A_{i}$, for example by setting $\phi=\sum_{i=1}^{n}\alpha_{i}\chi_{A_{i}}$.
A simple example of a simple-function is the indicator function $\chi_{A}$ of any $A\subset X$: it takes the value $1$ in $A$ and $0$ in the complement $A^{c}$.
The significance of simple-functions is that the measure-integral is defined through them. In fact, one can show that for any non-negative measurable function $f$ there exists a nondecreasing sequence of simple functions $(\phi_{i})$ so that $\phi_{i}\to f$ point-wise.