Measure Theory/Length-measure Integration is the One that We Want

Length-measure Integration Is the One that We Want

In a sense, this section finalizes what we set out to do: Obtain a notion of integration which can easily justify the interchange of limit and integral.

In the previous section we finally laid down all of the definitions needed, and we now know what the integral is. All that remains is to prove that it does what we need.

Of course, we don't just want the interchange of limit and integral. We also need to know that ... you know ... this thing is an integral. It should measure the area between a curve and the x-axis. It should distribute over summation and scalar multiples. It should split regions in the sense that

$\int _{[a,b]}f=\int _{[a,c]}f+\int _{[c,b]}f$

You know, all the stuff we have come to love and expect from an integral.

Length-measure and Riemann Integrals

A major result that we will show is that the length-measure integral and the Riemann integral "agree" in a large class of cases, especially ones in which we understand what the area under the curve should be. This will assure us that length-measure is a reasonable formalization of area under a curve. Along the way we will prove that many of the familiar properties of integration, mentioned above, also hold for length-measure integration.

Convergence Theorems

The final payoff will be when we show the convergence theorems.

Bounded Integrals

• Bounded Convergence Theorem

Nonnegative Integrals

• Fatou's Lemma
• Monotone Convergence Theorem

General Integrals

• Lebesgue Convergence Theorem
• Generalized Lebesgue Convergence Theorem

Convergence in Measure

Finally, we briefly introduce the concept of convergence in measure. This is of particular importance when measure theory is applied to probability theory.

The various convergence theorems all continue to hold, when "convergence" is replaced by "convergence in measure".