Approximation by Step Functions

It turns out you can approximate functions by step functions. Let’s make this rigorous with measure theory. These are my notes of chapter 3B of Axler’s MIRA.

Notation

We refer to the space of absolutely integrable real functions under the standard Lebesgue measure $\lambda$ by $\mathcal{L}^1(\textbf{R})$. The norm $\|f\|_1$ denotes the integral of the abosulte value of $f$ with respect to $\lambda$:

$$\htmlId{eq-1}{\|f\|_1 = \int |f| d \lambda.} \tag{1}$$

Step Functions

A step function is a function that takes on finitely many values, each on a union of bounded intervals of the real line. Formally:

These will prove to be useful tools when discussion the space $\mathcal{L}^1(\mathbf{R})$ because integrals of step functions are straight-forward. If $g$ is a step function, as above, and the intervals $I_1, \dotsc, I_n$ are disjoint, then

$$\htmlId{eq-3}{\| g\|_1 = |a_1| |I_1| + \cdots + |a_n| |I_n|.} \tag{3}$$

In particular, $g \in \mathcal{L}^1(\mathbf{R})$ if, and only if, all the intervals $I_1, \dotsc, I_n$ are bounded.

We can approximate functions in $\mathcal{L}^1(\mathbf{R})$ by step functions.

While this is nice, it really is just a necessary step in proving the following theorem. Recall Luzin’s theorem, which gives a method of approximating a Borel measurable function by a continuous function. While spectacular, the following result is usually more useful.

The sketch of the proof is that, as above, we can approximate $f$ using step functions. And, for each of those steps, we can approximate it arbitrarily well with a continuous function that looks like a trapezoid. Then, $g$ is the sum of all those continuous functions, and the triangle inequality bounds the total error.