Regularity, the degree of differentiability of a smooth function
Seems to make sense in the context you just described.
I often read the term regularity conditions in different contexts. But what does this mean exactly.
For example I read "The Lebesgue theory requires a very weak regularity condition called measurability".
Now I know the Lebesgue integral is only defined for measurable functions. So i understand what the author is trying to say. But in general what are regularity conditions?
Nah, typically regularity conditions don't refer to that. Many measurable functions wouldn't qualify as having any regularity at all under that definition. A regularity condition is essentially just a requirement that whatever structure you are studying isn't too poorly behaved. For instance, in the context of Lebesgue integration, the existence of a dominating function would be considered a regularity condition required to carry out various limit interchanging processes. In probability theory, on the other hand, the existence of a finite second moment would be considered a regularity condition for applying the Central Limit Theorem.