I read the following paragraph from wikipedia:
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the $L^1$ norm. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration.
Can any one explain why the power of this theorem is the primary theoretical advantage of Lebesgue integration over Riemann? It would be great if you could illustrate with examples. Thanks!
Best Answer
Here is an exercise from Rudin's "Real and Complex Analysis" which I think shows the point:
The solution involving the Dominated Convergence Theorem is as follows:
Notice the functions $f_n$ are dominated by the constant $1$ function, which is $L^1$ since $[0,1]$ has finite measure. Then we see
$$ \lim_{n \to \infty} \int_0^1 f_n(x)~d\mu \overset{(1)}{=} \int_0^1 \lim_{n \to \infty} f_n(x)~d\mu \overset{(2)}{=} \int_0^1 0~d\mu = 0 $$
where (1) follows from the DCT and (2) follows from the assumption $f_n(x) \to 0$. If you want to be extremely precise, you can then note that the $f_n$ and $0$ are all riemann integrable too. Since lebesgue integration and riemann integration coincide for riemann integrable functions, this means we have also shown the claim with $d\mu$ replaced by $dx$ above... though I feel this is a bit of a pedantic note.
The important part of the exercise, though, is the following: Try to prove this theorem without the machinery of measure theory. I certainly don't know how I would do it, and that's the point.
In general, the lebesgue integral works better with respect to limits because of the Dominated Convergence Theorem and its sister the Monotone Convergence Theorem. Since we are often interested in taking limits, this is obviously a benefit. In fact, this benefit is even larger than it initially seems:
The space of Riemann integrable functions on $[0,1]$, with its natural inner product
$$\langle f, g \rangle = \int_0^1 f(x) \cdot g(x) dx$$
is not complete, because there are Cauchy sequences of riemann integrable functions whose limit is not riemann integrable (exercise :P). The lebesgue integral solves this problem (using the DCT), allowing for the use of Banach and Hilbert Space theory.
I hope this helps ^_^