External (outer) measure

This video defines the concept of external measure (outer measure) and proves some basic properties. The external measure plays the fundamental role in the Lebesgue measure extension process. The most required property of the external measure is semi-additivity.

A good example, where the outer measure construction is highly non-trivial is the example of "length" measure I discuss here:

[durackpl]

That is true! A good example of semi-ring were this works, is the semi-ring of half-open intervals. That is, the semi-ring you use to build the Lebesgue measure on the line, the "length" measure.

I discussed "length" measure in my other comments. The link is in the description above.

[durackpl]

This is a rather exreme case, but I think you are correct, if there is not covering, then you cannot define the outer measure. On the other hand, the outer measure is specifically designed for Lebesgue extension, which is done in "finite case". That is, the case when the semi-ring contains the universal set X, so any subset in fact does have a covering.

[endelecheia]

I'm a little confused by the definition of an external measure. What if there isn't a covering of the set A by elements of the semi-ring S? Does that mean that we cannot define an external measure for the set A? In other words, what guarantees the existence of a covering of A?

[endelecheia]

So, if I understand you correctly, we have to choose an appropriate semi-ring, i.e. one that can be used to construct coverings for all subsets of X? Is there any example that illustrates the idea? Thanks!

[mathematicsbyahsanmohsan]

please tell me how i can download this lecture

[luozhiyuan]

At time 11:23, I think m*(A) in the right side of inequality should be m*(B).