# The Unapologetic Mathematician

## Simple and Elementary Functions

We now introduce two classes of functions that are very easy to work with. As usual, we’re working in some measurable space $(X,\mathcal{S})$.

First, we have the “simple functions”. Such a function is described by picking a finite number of pairwise disjoint measurable sets $\{E_i\}_{i=1}^n\subseteq\mathcal{S}$ and a corresponding set of finite real numbers $\alpha_i$. We use these to define a function by declaring $f(x)=\alpha_i$ if $x\in E_i$, and $f(x)=0$ if $x$ is in none of the $E_i$. The very simplet example is the characteristic function $\chi_E$ of a measurable function $E$. Any other simple function can be written as

$\displaystyle f(x)=\sum\limits_{i=1}^n\alpha_i\chi_{E_i}(x)$

Any simple function is measurable, for the preimage $f^{-1}(A)$ is the union of all the $E_i$ corresponding to those $\alpha_i\in A$, and is thus measurable.

It’s straightforward to verify that the product and sum of any two simple functions is itself a simple function — given functions $f=\sum\alpha_i\chi_{E_i}$ and $g=\sum\beta_j\chi_{F_j}$, we have $fg=\sum\alpha_i\beta_j\chi_{E_i\cap F_j}$ and $f+g=\sum(\alpha_i+\beta_j)\chi_{E_i\cap F_j}$. It’s even easier to see that any scalar multiple of a simple function is simple — $cf=\sum c\alpha_i\chi_{E_i}$. And thus the collection of simple functions forms a subalgebra of the algebra of measurable functions.

“Elementary functions” are similar to simple functions. We slightly relax the conditions by allowing a countably infinite number of measurable sets $E_i$ and corresponding values $\alpha_i$.

Now, why do we care about simple functions? As it happens, every measurable function can be approximated by simple functions! That is, given any measurable function $f$ we can find a sequence $f_n$ of simple functions converging pointwise to $f$.

To see this, first break $f$ up into its positive and negative parts $f^+$ and $f^-$. If we can approximate any nonnegative measurable function by a pointwise-increasing sequence of nonnegative simple functions, then we can approximate each of $f^+$ and $f^-$, and the difference of these series approximates $f$. So, without loss of generality, we will assume that $f$ is nonnegative.

Okay, so here’s how we’ll define the simple functions $f_n$:

\displaystyle f_n(x)=\left\{\begin{aligned}\frac{i-1}{2^n}\qquad&\frac{i-1}{2^n}\leq f(x)<\frac{i}{2^n},\quad i=1,\dots,n2^n\\n\qquad&n\leq f(x)\end{aligned}\right.

That is, to define $f_n$ we chop up the nonnegative real numbers $\left[0,n\right)$ into $n2^n$ chunks of width $2^n$, and within each of these slices we round values of $f$ down to the lower endpoint. If $f(x)\geq n$, we round all the way down to $n$. There can only ever be $n2^n+1$ values for $f_n$, and each of these corresponds to a measurable set. The value $\frac{i-1}{2^n}$ corresponds to the set

$\displaystyle f^{-1}\left(\left[\frac{i-1}{2^n},\frac{i}{2^n}\right)\right)$

while the value $n$ corresponds to the set $f^{-1}\left(\left[n,\infty\right]\right)$. And thus $f_n$ is indeed a simple function.

So, does the sequence $\{f_n\}$ converge pointwise to $f$? Well, if $f(x)=\infty$, then $f_n(x)=n$ for all $n$. On the other hand, if $k\leq f(x) then $f_k(x)=k$; after this point, $f_n(x)$ and $f(x)$ are both within a slice of width $\frac{1}{2^n}$, and so $0\leq f(x)-f_n(x)<\frac{1}{2^n}$. And so given a large enough $n$ we can bring $f_n(x)$ within any desired bound of $f(x)$. Thus the sequence $\{f_n\}$ increases pointwise to the function $f$.

But that’s not all! If $f$ is bounded above by some integer $N$, the sequence $f_n$ converges uniformly to $f$. Indeed, once we get to $n\geq N$, we cannot have $f_n(x)=n$ for any $x\in X$. That is, for sufficiently large $n$ we always have $0\leq f(x)-f_n(x)<\frac{1}{2^n}$. Given an $\epsilon>0$ we pick an $n$ so that both $n\geq N$ and $\frac{1}{2^n}<\epsilon$, and this $n$ will guarantee $\lvert f(x)-f_n(x)\rvert<\epsilon$ for every $x\in X$. That is: the convergence is uniform.

This is also where elementary functions come in handy. If we’re allowed to use a countably infinite number of values, we can get uniform convergence without having to ask that $f$ be bounded. Indeed, instead of defining $f_n(x)=n$ for $f(x)\geq n$, just chop up all positive values into slices of width $\frac{1}{2^n}$. There are only a countably infinite number of such slices, and so the resulting function $f_n$ is elementary, if not quite simple.

May 11, 2010 - Posted by | Analysis, Measure Theory

## 12 Comments »

1. […] We start our turn from measure in the abstract to applying it to integration, and we start with simple functions. In fact, we start a bit further back than that even; the simple functions are exactly the finite […]

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2. […] of Integrable Simple Functions We want to nail down a few basic properties of integrable simple functions. We define two simple functions and to work […]

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3. […] Properties of Integrals Today we will show more properties of integrals of simple functions. But the neat thing is that they will follow from the last two properties we showed yesterday. And […]

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4. […] our recent results, today’s proposition is specifically stated and proved for integrable simple functions, and won’t be generalized […]

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5. […] to be integrable: a function is integrable if there is a mean Cauchy sequence of integrable simple functions which converges in measure to . We then define the integral of to be the […]

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6. […] before, but now for general integrable functions. Similarly, if is nonnegative a.e., we can find a sequence of nonnegative simple functions converging a.e. (and thus in measure) to . The integral […]

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7. […] is simple, this is obvious. Indeed, if is the collection of sets used to write as a finite linear […]

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8. […] case, will be integrable with respect to both and . Indeed, since this is obviously true for simple , and general integrable functions are limits of simple integrable […]

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9. […] let be an increasing sequence of non-negative simple functions converging pointwise to . Then monotone convergence tells us […]

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10. […] if is a non-negative simple function, then we can write it as the linear combination of characteristic functions of disjoint subsets. If […]

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11. […] will be sufficient to establish this for simple functions, since for either the upper or the lower ordinate set we can approximate any measurable by a […]

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12. […] we assume that is a simple function. Then is a finite linear combination of characteristic functions of measurable sets. But clearly […]

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