nLab simple function

Idea

Simple functions are (almost) the most basic notion of measurable function in measure theory. Given a measure, it's easy to define the integral of a simple function, and we extend this to more general functions by continuity.

Definitions

Let $X$ be a measurable space. We may want $X$ to be equipped with some more data; if $X$ is a measure space, then this is plenty of data. However, for the most basic definitions, it's enough if $X$ is simply a measurable space. This is the domain of our simple functions.

Another necessary datum is the simple functions' codomain $K$, which we will eventually want to be at least a Banach space over the real numbers. (In the simplest example, $K$ is $\mathbb{R}$ itself, or perhaps the space $\mathbb{C}$ of complex numbers.) We take $K$ to be a measurable space using its Borel sets.

Definition

A measurable function from $X$ to $K$ is simple if its range is finite.

Since a simple function $f$ is measurable and a singleton is Borel, each fibre of $f$ is a measurable set in $X$; the function $f$ is given by the (finitely many) nonempty fibres and their (singleton) images. This suggests another way to look at simple functions:

Definition

A simple function from $X$ to $K$ is a formal $K$-linear combination of measurable subsets of $X$.

Here we identify a measurable set $A$ with its characteristic function $\chi_A$, so the formal linear combination $\sum_i c_i A_i$ is identified with the function $\sum_i c_i \chi_{A_i}$, which is measurable and whose range is contained in the finite set of sums of the $c_i$. (If there are $n$ terms in the linear combination, then there are at most $2^n$ such sums.)

However, the naïve partition by equality induced by linear combinations is finer than equality of the corresponding functions, so we must combine Definition with a definition of equality:

Definition

Two simple functions from $X$ to $K$, in the sense of Definition , are equal if their corresponding functions from $X$ to $K$ are equal as functions.

Then we have a canonical bijection between the set of simple functions as in Definition and the set of equivalence classes of simple functions as in Definition .

Arguably, even this is not really the correct notion of equality, since functions may be equal for the purpose of integration without being literally equal. If $X$ is equipped with a $\sigma$-ideal of null sets (or a $\delta$-filter of full sets), then we may consider a yet coarser notion of equality:

Definition

Two simple functions, in the sense of either Definition or Definition , are almost equal if they (or their corresponding functions) are equal almost everywhere.

Sometimes, we wish to restrict attention to those simple functions which we expect to have a finite integral. If $X$ is equipped with an ideal of bounded sets (which in a measure space are sets with finite measure), then we may do this:

Definition

A simple function of bounded support is a simple function in the sense of Definition such that the fibre over every non-zero number is bounded, or equivalently (in the sense of Definition ) a formal linear combination of bounded measurable sets.

In some approaches to measure theory, one starts with a $\delta$-ring of measurable sets, which may be reinterpreted as the bounded sets in the generated $\sigma$-algebra of relatively measurable sets, and then the simple functions will automatically have bounded support.

Finally, there is one more useful restriction (and slight generalisation) of simple functions, applicable when $K$ is ordered:

Definition

A positive simple function is a simple function in the sense of Definition whose range is contained in the positive cone $K^+$ of $K$, or equivalently (in the sense of Definition ) a formal $K^+$-linear combination of measurable sets. An extended positive simple function (note the red herring) takes values in the extended positive cone $\bar{K}^+$, or equivalently is a $\bar{K}^+$-linear combination.

Integration

Let $X$ be equipped with a measure $\mu$, so $(X,\mu)$ is a measure space. (In particular, $X$ has the structure necessary for all of the definitions above, including both Definitions and .)

If $f$ is a simple function from $X$ to $K$, then we wish to define the integral of $f$. In general, this is a little tricky, but it's easy if $f$ either is positive or has bounded support. It is easiest to write down the definition if we think of simple functions using Definition . Then we have:

Definition

The integral of the simple function $f$, represented by the linear combination $\sum_i c_i A_i$, is $\sum_i c_i \mu(A_i)$.

Proposition

The integral of a positive simple function always exists (but may be infinite). It is finite if $\mu$ is a finite measure, and it is positive (possibly $0$ or $\infty$) if $\mu$ is a positive measure. Also, if $\mu$ is positive, then the integral of an extended positive simple function always exists.

(However, the integral of an extended positive simple function with respect to a finite positive measure need not be finite.)

Proposition

The integral of a simple function with bounded support always exists and is finite (being a finite linear combination of finite numbers).

Proposition

Two (positive or with bounded support) simple functions $f$ and $g$ are almost equal (with respect to $\mu$) if and only if the integral of $f - g$ is zero.

Definition

The $L^1$-norm of a simple function is the integral of its pointwise norm (which is a positive simple function to $\mathbb{R}$) with respect to the absolute value of the measure $\mu$ (which is a positive measure):

$${\|f\|}_1 \coloneqq \int {\|f(x)\|} {|\mu(\mathrm{d}x)|} .$$

In this context, we usually start with a positive measure $\mu$; in that case, of course, there is no need to bother taking the absolute value of $\mu$.

Proposition

The simple functions of bounded support form a normed vector space $Simp_c$ under the $L^1$-norm, if we consider them up to almost equality.

If we don't use almost equality, then we get in general only a seminorm, but if we pass to a quotient space with a norm, then Proposition tells us that we are now using almost equality (and shows that Definition is well defined when applied to Definition ).

Definition

The completion of the normed vector space $Simp_c$ (under the $L^1$-norm) is the Banach space $L^1$ of absolutely integrable functions (an example of a Lebesgue space).

Proposition

Taking the integral of a simple function of bounded support is a continuous linear functional on $Simp_c$, so it extends to all of $L^1$.

In this way, we may define the integral of any absolutely integrable function.

There might be some technical requirements for this to be true. I'll try to check on that.