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.