Why Lebesgue Integration

Tries to motivate Lebesgue Integration by trying to find an improved limit theorem for Riemann Integration.

In the process, find that this seems a lot like trying to find limit results when only dealing with rational numbers.

End up with a example where we can "see" what the integral is, but Riemann does not have an answer.

From the example see that by describing the unusual function by its range rather than its domain, we can compute the integral

Problem is that, in general we need to find the length of complicated sets

Outer measure is introduced to do just that.

Unfortunately, this doesn't always work.

Define a subset of all sets called measurable sets.

These are the building blocks that can be used to create "simple" functions

And these can be used to define any function that is a point-wi ...