I've lately read multiple times the term "Lebesgue representative" and wanted to ask what this actually precisely stands for? I guess this refers to the Lebesgue differentiation theorem, right? So it basically is the $L^p$-function's "value" when letting the radius over a ball of a mean value integral tend to zero, right?
Thanks!
In the context of Lebesgue integration, there is a very useful equivalence relation between measurable functions: two functions are equivalent iff they are equal almost everywhere.
Two functions which are equal almost everywhere will be integrable on the same measurable subsets, and will have exactly the same integral on any given measurable subset.
If you only care about integrals $\int_A f d\mu$, and not about values $f(x)$, then it's often interesting to work on the quotient space, in which two functions which are equal almost everywhere correspond to the same element. Formally, the quotient space is the space of classes of equivalence.
However, the elements of this quotient space are rather abstract objects; the easiest way to describe an element of the quotient space, that is, an equivalence class, is to give one function in that class. This function is called a representative of the class.
This is not very different to working with the quotient set $\mathbb{Z}/n\mathbb{Z}$, in which elements are classes of equivalence of integers, and each equivalence class can be represented by an integer from that class.