Littlewood's three principles

1. Nov 18, 2009

guildmage

How do you explain Littlewood's three principles in simpler terms? What does "nearly" mean (as in nearly a finite union of intervals, nearly continuous, and nearly uniformly convergent)?

And why are these important if I'm going to study the Lebesgue integral?

I'm learning this on my own so I'm really having a hard time digesting what the book (Real Analysis by Royden) is saying.

2. Nov 18, 2009

Hurkyl

Staff Emeritus
Those are the simpler terms. "Nearly" is purposely vague; these are approaches to problem solving that you're supposed to adapt to the problem of interest. Several commonly useful versions appear in the chapter; the introductory paragraphs to that section names them.

3. Nov 18, 2009

fourier jr

"nearly" is another way of saying "almost everywhere" isn't it?

4. Nov 19, 2009

guildmage

@fourier jr: I would like to believe so.

5. Nov 19, 2009

g_edgar

No.

One principle says: A set is nearly a finite union of intervals.

It does not mean "almost everywhere".

Another principle says: A function is nearly continuous. Precise meaning: see Lusin's Theorem. Again, it does not mean "almost everywhere".

The third is about uniform convergence. Precise meaning: Egorov's Theorem.

6. Nov 20, 2009

fourier jr

i've even got that book by royden. i guess i haven't looked at it in a while :(