For me the trouble with Lebesgue measure is that it is quite specialized.

In probability we might try to measure the probability of an event using other observations at our disposal. Hopefully some combination of unions and intersections, however complicated might give us a meaningful result.

However, there are cases where we can't measure given the information. We can't measure temperature with a ruler.

Maybe... We can use the expansion or contraction with respect to temperature. Then given a prior series of observations we might try to combine them to deduce temperature from length measurements.

For me the trouble with Lebesgue measure is that it is quite specialized.

In probability we might try to measure the probability of an event using other observations at our disposal. Hopefully some combination of unions and intersections, however complicated might give us a meaningful result.

However, there are cases where we can't measure given the information. We can't measure temperature with a ruler.

Maybe... We can use the expansion or contraction with respect to temperature. Then given a prior series of observations we might try to combine them to deduce temperature from length measurements.

If we solve a partial differential equation using Fourier series. We might try to study how much error we created by truncating the series after finitely many terms. The nature of the error might be studied with Lebesgue measure and Lebesgue integral.

Another consideration is a very complicated set. In principle any measurable set will be covered with squares or rectangles or circles. However, we might be limited by how small the circles are, or how many circles we can use, or be forced to use squares instead of circles.

An example, what is the minimum number of squares to cover a circle with 1% error?