home *** CD-ROM | disk | FTP | other *** search
- Newsgroups: sci.math
- Path: sparky!uunet!snorkelwacker.mit.edu!galois!riesz!tycchow
- From: tycchow@riesz.mit.edu (Timothy Y. Chow)
- Subject: Dominated Convergence Theorem
- Message-ID: <1992Sep10.015756.17302@galois.mit.edu>
- Sender: news@galois.mit.edu
- Nntp-Posting-Host: riesz
- Organization: None. This saves me from writing a disclaimer.
- Date: Thu, 10 Sep 92 01:57:56 GMT
- Lines: 16
-
- The DCT for real measurable functions states that if f_n -> f and there
- exists integrable g such that |f_n| <= g for all n, then the limit of the
- integrals equals the integral of the limit.
-
- Here's a proof of DCT using Fatou's lemma.
- 1. Reduce to the nonnegative case by adding g to everything.
- 2. Applying Fatou to f_n implies integral(f) <= liminf integral(f_n).
- 3. Applying Fatou to g-f_n implies integral(f) >= limsup integral(f_n).
-
- I haven't seen this proof in any book. Why not? It seems very simple and
- intuitive.
- --
- Tim Chow tycchow@math.mit.edu
- Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
- 30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
- only 1 1/2 tons. ---Popular Mechanics, March 1949
-