Approximating x^2 as a sequence of simple functions

[Yagoda]

Homework Statement

I'm looking for a sequence of simple functions f_n that converges uniformly to f(x)=x² on the interval [0,1].

Homework Equations

I know a simple function is one that can be written as $\sum^{n}_{k=1}a_{k}1_{D_{k}}(x)$ where {D₁,...,D_n} is collection of measurable sets and a₁,...,a_n are real numbers.

The Attempt at a Solution

I don't really know how to begin here. My general idea would be to break up the range of the function into pieces, take the inverse image of these pieces and make these the D's, the measurable sets whose characteristic functions we are looking at. The a's, the actual values that the simple function takes on, could be the smallest value in a particular piece of the range. The higher terms of the sequence would break the range into smaller and smaller pieces.

But how would I actually write down this sequence of functions?

 

[Dick]

Homework Statement

I'm looking for a sequence of simple functions f_n that converges uniformly to f(x)=x² on the interval [0,1].

Homework Equations

I know a simple function is one that can be written as $\sum^{n}_{k=1}a_{k}1_{D_{k}}(x)$ where {D₁,...,D_n} is collection of measurable sets and a₁,...,a_n are real numbers.

The Attempt at a Solution

I don't really know how to begin here. My general idea would be to break up the range of the function into pieces, take the inverse image of these pieces and make these the D's, the measurable sets whose characteristic functions we are looking at. The a's, the actual values that the simple function takes on, could be the smallest value in a particular piece of the range. The higher terms of the sequence would break the range into smaller and smaller pieces.

But how would I actually write down this sequence of functions?

I would pick your measurable sets to be n intervals of size 1/n that cover [0,1]. Then you just have to define a_n for each interval. Don't get intimidated by the terminology. It's really not that hard.

 

[Yagoda]

You're right that it actually wasn't that difficult. I was still trying to wrap my head around simple functions. Thanks for your help!