In one book I saw that they constructed the lebesgue measure via semialgebras etc, starting with the intervals $(a,b]$, a and b is finite, and $(-\infty, b]$, and $(a,\infty)$.
But please take a look at this wikipedia article.: http://en.wikipedia.org/wiki/Lebesgue_measure#Definition
Here they say they construct it using all closed, open and semiclosed intervals, I assume they are $(a,b),(a,b],[a,b),[a,b]$. But is infinity included here? Or will you get the measure without thinking about infinity?
They do something of the same in a new book I am reading. They start defining a set with all type of intervals. But they do not say if infinity can be one of the endpoints. But then they start to talk about functions that are finite on the set. And the length function is such a function, and so this "says" that we only look at final intervals?
Basically, if you are constructing the Lebesgue measure, and you look at the 4 intervals open, closed and semi-closed, is it implicitly assumed that all of them are finite?
UPDATE: Was asked about what book I used in the comments, it was Principles of mathematical analysis 3rd. edition, by rudin. Here is a screenshot:
