My question involves the definition of the Lebesgue integral. Most colloquial definitions I've read follow (2), in that f*(t) is the "length" of one of the horizontal rectangles and dt is the height. But, I've also seen definitions follow (1), which appear fundamentally different. Why are they presented differently? Also, are the two forms somehow equivalent to each other? Is there a good way to visualize why they are equivalent? Why are they presented differently?
My next question is the notation in (3). This notation is said to be the Lebesgue integral in it's "standard" form. My interpretation of this integral is that it is integrated over E, which is the domain of the function f. This seems more akin to Riemann integration where the domain is partitioned rather than the range (in this case if the measure space is (E,X,μ) and f : E->R, where R is the reals). Shouldn't it be summed over R instead? also how should dμ be interpreted? If you look at (4) it makes sense. The integral is integrated over the domain of f, E. Then, the function is evaluated at x ϵ E. μ(dx) also makes sense, as dx is some segment of E, which is an element of the sigma space X. The μ(dx) is then the measure of this element of the sigma space. But, as I mentioned before, this seems to just be Riemann integration.
In summation, which (1) or (2) better describes Lebesgue integration and why are these different approaches use to describe Lebesgue integration; how should the notation in (3) be interpreted. Thanks.