what is the intuition of a "measureable function"?

Question

I am trying to learn enough analysis to underpin some complex probability theory. It always comes back to measure theory, of course.

I get the idea of measure in the sense that the measure of the function $f(x)=1$ when $x$ is rational, $0$ otherwise has measure zero on the interval $[0,1]$, whereas the function $f(x)=1$ over that same interval has measure $1$.

But what does it mean for a function to be measureable? Is there some straightforward intuition that does not require pathological cases like the function $f(x)$ defined above?

Answer 1

A function $f$ is measurable if the preimages of intervals under $f$ are measurable sets. This is precisely what is required in order to attempt to define its Lebesgue integral. I use the word "attempt" because not all measurable functions are integrable due to issues pertaining to infinities.

Going in the direction of intuition, in analysis, intuitively all functions are measurable, because we use huge $\sigma$-algebras on the domain. This intuition is not quite correct; using non-constructive methods such as the axiom of choice, one can "construct" Lebesgue nonmeasurable sets and thus Lebesgue nonmeasurable functions. But there are precise senses in which these pathological situations "never happen".

In probability theory the situation is a bit different. In probability theory we still work with a large "base" $\sigma$-algebra where measurability is generally no concern. (I can only think of one exception to this statement: because of measurability issues, it is impossible to have a continuum of iid nondegenerate Gaussians, which is exactly what one would like to have in order to define white noise as a stochastic process in the strict sense.)

But we also work with sub-$\sigma$-algebras, with respect to which there are plenty of nonmeasurable functions. These sub-$\sigma$-algebras are interpreted as information: the events in them are the events with the property that we can know whether or not they occurred after being given only some incomplete information about what occurred (for example, the first few values of a discrete time stochastic process).

Answer 2

Have you checked out the Wikipedia page on measure? I think it explains the basics pretty well. An intuitive interpretation is that the measure of a set is supposed to represent its size. So for example, the measure of an interval $(a,b) \subset \mathbb{R}$ is $b-a$.

In the example you gave, the rational numbers have measure $0$ in the reals. The connection to probability might supplement your intuition: what's the probability that a random $x\in[0,1]$ is rational?

Answer 3

Suppose someone is flipping coins: every minute from time $1$ onwards, they flip one coin. The result of their coin flips is recorded in a book.

An event will be a YES / NO question about the coin flips. An event will be measurable at time $N$ if we can answer YES or NO based on the knowledge available to us at time $N$.

At time $N$, only certain events are measurable. For example, we can determine if the number of heads at time $N$ exceeded $3$, or if more than half of the flips were heads.

Some events are not measurable at time $N$. For example, we don't know what the outcome of the $N+1$st coin flip will be, so we can't answer YES or NO to is the $N+1$st coin heads?

Formally, at time $N$ there are $N$ functions we have access to, namely the first $N$ coin flips: $X_1, \ldots, X_N$. $X_1, \ldots, X_n$ generate a sigma algebra, $\sigma(X_1, \ldots, X_N)$, which is the smallest sigma algebra that contains the events $\{X_i = H\}$, $i = 1, \ldots N$.

Given knowledge at time $N$ (and a idealized computer) we can evaluate any function $f(X_1. \ldots, X_N)$, where $f(x_1, \ldots, x_N)$ is some deterministic function of $n$ variables. The functions $f(X_1, \ldots, X_N)$ are functions on the probability space* underlying the experiment of coin flips, and they are exactly functions which are measurable with respect to the sigma algebra $\sigma(X_1, \ldots, X_N)$.

*Note that this probability space usually comes with the sigma algebra $\mathscr{F}$ generated by the events $\{X_i = H\}$ for all $i$. Any function which we can "compute" (assuming an idealized computer) given knowledge of all coin flips is measurable with respect to $\mathscr{F}$.

Given any measurable function $f$, the event $\{ f < \alpha \}$ is measurable. The intuition is that we can use our data to "measure" whether or not this event is true. Our "data" or "knowledge" is encoded in a sigma algebra, which is generated by the basic questions we can ask, e.g. "is the 10th coin heads?"