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I'm just reading Rudin's Principles of mathematical analysis - the last chapter on Lebesgue integration and I am having a bit trouble understanding the motivation of the definition of Lebesgue measure.

This is how I understand it:

We want to measure sets in [tex]\mathds{R}^n[/tex] so what we have to do is to find some [tex]\sigma[/tex]-algebra on [tex]\mathds{R}^n[/tex] and to define measure on the [tex]\sigma[/tex]-algebra. Now the sets, which we want to have in the [tex]\sigma[/tex]-algebra are mainly intervals and their countable unions.

So we seek, and find out, that there exists such a [tex]\sigma[/tex]-algebra (denoted by- [tex]\mathfrak{M} (\mu)[/tex]) consisting of so called [tex]\mu[/tex]-measurable sets and there also exists a regular, countably additive, nonnegative (did I forget something?) set function [tex]\mu[/tex].

Now my questions are:

1. Is there some "larger" [tex]\sigma[/tex]-algebra containing [tex]\mathfrak{M} (\mu)[/tex] or is [tex]\mathfrak{M} (\mu)[/tex] the largest?

2. Does [tex]\mu[/tex] have to be regular on [tex]\mathfrak{M} (\mu)[/tex]?