Uniqueness of Weak Partial Derivatives

Question

I have a quick question about a theorem:

Suppose that the weak derivative of $u$ with order $\alpha$ exists. Then it is uniquely defined up to a set of measure zero.

What does it mean for something to be defined up to a set of measure zero? Is it just a synonym for almost everywhere?

Answer 1

you're right. If two functions are equal up to a set of measure zero, we say that they are equal almost everywhere (a.e.).