Approximate $f$ by simple functions

Question

Consider the measure space $([0,1)^2, \mathcal{B}([0,1)^2, \lambda)$ where $\lambda$ is the Lebesgue measure on $[0,1)^2$. Put $f:[0,1)^2 \to \mathbb{R}, \,\,\, f(x,y) = x + 2y$.

I would like to approximate $f$ by simple functions in order to evaluate the integral $$\int_{[0,1)^2} f \, d\lambda$$

So basically I'm looking to approximate $f$ by a sequence $(f_k)_{k\in \mathbb{N}}$ with $0\leq f_k \leq f_{k+1}$.

I've tried splitting up the unit square into $2^{2k}$ squares and then defining a simple function $f_k$ using the minimum that $f$ attains on each of the $2^{2k}$ squares. However, this construction is a lot of effort to write down explicitly and I'm thinking there must be a better way (but I can't think of it).

So please give me a hint to get on the right track.

Answer 1

Your idea is correct. And as the commenters on your question have said, the use of simple functions is mostly theoretical. Here is the function that I think you have meant (or at least along the same line of thought):

$$ f_n = \sum_{i=1}^n \sum_{j=1}^n \left(\frac{i-1}{n} + 2\frac{j-1}{n} \right) \times 1\!\!1_{[\frac{i-1}{n},\frac in]\times[\frac{j-1}{n},\frac jn]} $$

Where $1\!\!1_{[a,b]\times[c,d]}$ is an indicator function of a rectangle in $\Bbb R^2$. This task was simplified by the fact that your function is linear, and as such attains its minimal value on the border (the lower left corner of each rectangle), which is the value in brackets in the sum.

Answer 2

Bartle uses the following:

Let $n$ be a fixed natural number. If $k=0,1,..., n2^n-1$, let $E_{kn}$ be the set $$E_{kn}=\{x\in X:k2^{-n}\leqslant f(x)\leq(k+1)2^{-n}\},$$ and if $k=n2^n$, let $E_{kn}$ be the set $\{x\in X: f(x)\geqslant n\}$. Then define $\phi_n$ to be equal to $k2^{-n}$ on $E_{kn}$