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

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In summary, Littlewood's three principles are approaches to problem solving that are useful when studying the Lebesgue integral, as explained in Royden's Real Analysis. These principles include the idea of a set being nearly a finite union of intervals, a function being nearly continuous (as defined by Lusin's Theorem), and the concept of uniform convergence (as defined by Egorov's Theorem). These principles are not the same as "almost everywhere" and are important to understand when studying the Lebesgue integral.
  • #1
guildmage
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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.
 
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  • #2
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
"nearly" is another way of saying "almost everywhere" isn't it?
 
  • #4
@fourier jr: I would like to believe so.
 
  • #5
fourier jr said:
"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.
 
  • #6
i've even got that book by royden. i guess i haven't looked at it in a while :(
 

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

What are Littlewood's three principles?

Littlewood's three principles are a set of guidelines proposed by mathematician John Edensor Littlewood for determining the occurrence of unlikely events. These principles are often used in statistical analysis and probability theory.

What are the three principles?

The three principles are the Law of Truly Large Numbers, the Law of Near Enough, and the Law of Inevitable Surprises.

What is the Law of Truly Large Numbers?

The Law of Truly Large Numbers states that in a large enough sample size, even the most unlikely events will occur. This means that as the number of observations increases, the probability of rare events happening also increases.

What is the Law of Near Enough?

The Law of Near Enough states that events that are very close to happening or not happening should be treated as if they are certain to happen or not happen. This principle is used to simplify complex calculations and make them more manageable.

What is the Law of Inevitable Surprises?

The Law of Inevitable Surprises states that unlikely events will happen more frequently than we expect, and we should be prepared for them. This principle emphasizes the need for caution and preparedness, even in situations where the probability of a rare event occurring is low.

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