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Let me begin by providing the following two definitions from my class notes.

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I was trying to put together how from the definition of a simple function, we would form the given definition of an integral. Can someone help me put this together? Thanks.

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What do you mean by "form the definition"? You want to know the intuition behind the integral? – Michael Greinecker Feb 3 '14 at 1:22
@MichaelGreinecker Yes, that is what I meant. – Sarah Feb 3 '14 at 1:39
up vote 2 down vote accepted

Hopefully this will give some intuition:

Let's return to the idea that integration should represent area: Simple functions have graphs that look like rectangles (if the $A_i$ are rectangles, and a more complicated interpretation otherwise), so the integral should just be

$$(\text{height of $f$})(\text{width of $A_i$})$$

summed over all $i$. Now "width" means measure.

If one considers the first of Littlewood's Three Principles, this is the only natural thing to do. The Lebesgue integral can be viewed as the completion of the Riemann integral on a much larger class of functions (namely, the completion of the continuous, compactly supported functions on $\mathbb{R}$ or $\mathbb{C}$), so it needs to preserve the idea that integration has something to do with areas.

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The standard way to proceed from integration of simple functions to integration of positive functions is via a few results. These results should be present in your text, and so I am just providing the results, and a bit of "intuition".

1) Approximate a positive function defined on some set by simple functions: There are results that guarantee that such an approximation is possible.

2) Define the integral of the positive function to be the limit of the integrals of this sequence of simple functions.

Now you have to show that the result of integration doesn't depend on the sequence chosen.

Sometimes, just to avoid this problem, most texts do the following "trick" They define the integral of a positive function to be supremum of the following set: $\{\int{\phi}: \phi \leq f$, and $\phi$ is a simple function$\}$.

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