How to explain things in the motivation section of a mathematical paper without proper definitions?

When I was a Ph.D. student working on my own dissertation, I went to the university writing center for help and had a revelatory experience. The person working with me sat down with the first page of my introduction and effectively dissected it to identify the problems without understanding any of my technical jargon. They did this by reading aloud as we discussed, substituting blank/nonsense words for every piece of jargon, e.g.:

Here we apply method X to determine whether adjective thingies can be made to wibble.

This type of substitution forces you to step back from the technical world that you have dedicated so much time and love to, and understand your narrative---or lack thereof.

In your motivation, you need to take a couple of steps back and ask: why does anybody care about additive-finite measure space ("frobs") and how it relates to the space of mu-integrable functions ("greebit-space") or a stochastic process ("wibbling").

You didn't pick these elements at random. There must be some reason why you picked them and how they relate to the bigger community. Are they intended to solve a puzzle that a lot of people care about? Or a small piece of such a puzzle? Do they unite two sets of concepts that people thought were different? Will they help understand string theory or give better tools for interpreting MRI imaging?

You want to be able to write something like this:

People have wondered about how to better understand frobs ever since Richard Feynman first used them to pick the locks in Los Alamos. Although X, Y, and Z attempts have been made, none of them got very far because they were all green-colored. In this dissertation, I examine an alternate path, reducing the problem of frobs to the simpler system of greebit-space by means of an innovative application of wibbling. These results bring us one step closer to solving the problem of frobs, and how they can be better used to quickly and cheaply pick locks.

Now, what I've written is pure gibberish, and your motivation will almost certainly be much longer. The point, however, is this: your goal in a motivation section is to motivate by explaining that there is a problem that people care about and that you have an approach that gives at least a piece of the solution. Explain it in a way that your jargon can just be placeholders in the reader's mind, and it will be fine to leave the complex definitions for later.

When I encounter this problem, I write the introduction as if the readers knew the concepts that I mention, but I include a parenthetical comment or a footnote, after such a concept, along the lines of "This and other concepts used in the introduction will be defined in Section 2."

If you go deeply enough into measure theory and stochastic processes to actually write your dissertation about it, it is safe to assume that readers will be familiar with common concepts. So just assume that people understand what you write about. Do some handwaving if necessary ("we examine an interesting class of operators that are distinguished in that...").

Worry less about correctness than about telling a good story. After all, this is a motivational section. Don't include any definitions, or no more than one if it is utterly necessary. (And then, if you find that a definition is necessary in an introduction section, I'd argue that you probably need to revisit what you want to write in that section, until the definition is not necessary any more.)

Worry about correctness in the main body of your chapter.