When I read up on stochastic modeling, the use of "measure" comes up a lot. So far I just read the word "measure" as "probabilities" or "distribution" and was able to get away with it when trying to understand informally the concept and results of various models (Black, Dupire, Hull White..), at a basic level.
I must admit that I was not able to follow the rigorous proofs of every step in their derivation.
Can anyone tell me what is the significance of measure theory in stochastic modeling?
I do understand that the definition of a probability measure is more general and rigorous than just saying "probabilities". But apart from a rigorous definition, was there some useful results from measure theory that are often used in stochastic modeling?
Thanks very much and pardon my ignorance.