152k views
0 votes
Show that if a set T with m∗(T)=2023, then there is a bounded subset S of T such that m∗(S)>0.

User Nik Myers
by
8.8k points

1 Answer

5 votes

Final answer:

To show that if a set T with m*(T)=2023, then there is a bounded subset S of T such that m*(S)>0.

Step-by-step explanation:

We want to show that if a set T with m*(T)=2023, then there is a bounded subset S of T such that m*(S)>0.

In measure theory, the outer measure m*(T) of a set T is defined as the infimum of the sum of lengths of countable sequences of intervals that cover T. If m*(T)=2023, it means that there exists such a sequence of intervals whose sum of lengths is equal to 2023.

We can take any one of these intervals, and define S as the bounded subset that contains only that interval. Since the interval has a positive length, the measure of S will be greater than zero.

User Matthew Hui
by
7.4k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories