To be precise, I wonder if the term is used only for functions that is measurable with respect to the Lebesgue measure. Or is it used in general for any measurable function on a general measure space?

@ArcticChar I'm hoping to get some guidance on this too. The reason I added the tag was because one often encounters the functional in the question in a class dealing with Lebesgue integration.

Also, I'm wondering why real-analysis was inappropriate?

Looking at this discussion, it's a bit tricky for me to determine when a general tag should be avoided. E.g. I often find myself using stochastic-processes when dealing with martingales.

@JoseAvilez My take is that some tags are just too popular (like real-analysis, which is the most used tag) to be useful. Also, since measure theory is most appropriate there and is quite a popular tag. I consider that sufficient.

I don't know stochastic-process enough to say anything on that.