Understanding Littlewood's Three Principles in Relation to the Lebesgue Integral

[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.

 

[Hurkyl]

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.

 

[fourier jr]

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

 

[guildmage]

@fourier jr: I would like to believe so.

 

[g_edgar]

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

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.

 

[fourier jr]

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