So is there any consensus reached to refer to an integral with respect to a measure? I thought this type of integral is always called a Lebesgue integral. However, I noticed that some authors (such as Folland) use "Lebesgue integral" to refer to an integral with respect to the Lebesgue measure; he seems to just call an integral with respect to an arbitrary measure as an integral. I may be wrong about Folland's usage (which is not my point here :)!). But it does serve as an alert to be careful about the usage.
To be specific: is it clear enough, when there is no surrounding context, to refer to any integral with respect to a measure as a "Lebesgue integral"? I mean, would using "Lebesgue integral" lead the readers astray to think, "Okay, now we are thinking of an integral with respect to Lebesgue measure only.", in the absence of context. If not, what might be the better name?