Is it possible to do product measures before developing integration? I'm looking for an exposition to measure theory that defines integrals as the measure of the area under the curve. For this, one has to develop product measures before integration.
The only textbook that I know that does this is Pugh's Real Mathematical Analysis, but not in the generality that I'm interested in.
The MO thread Defining the integral of a function using the product measure seems to suggest that product measure without integration is outright impossible.
To clarify: I want to prove the existence of product measures while pretending that integration doesn't exist

Check the book Real Analysis - Modern Techniques and Their Applications, the author defines product measures without integration at all. 
In short, given two measurable spaces, first you define a rectangule as a cartesian product of two measurable sets. The disjoint union of all rectangules is an algebra, then you define a premeasure on this algebra as the product of the measure of the rectangules (the measure of each rectangule does not depend on integrals, only the very definition of measure). Finally, we have a theorem of extension which guarantes a sigma-algebra and a measure from this algebra with this premeasure (this theorem relies on outer measure, not integration at all). This is the product measure.
PS: it's true that the author makes use of integral to conclude how shoud be the premeasure, but this is more to motivate the definition. It's not a necessary part of the definitions and results.
