Prove if S and T are sets with outer content zero, SUT has outer content zero.

[STEMucator]

Homework Statement

Suppose that S and T are sets with outer content 0, prove that SUT also has outer content zero.

Homework Equations

C(S) denotes the outer content.

C(S) = C(T) = 0

Also : C(S) = inf \left\{{ \sum_{k=0}^{n} A_k}\right\} where A_k is the area of one of the sub-rectangles R_k.

The Attempt at a Solution

So we want to show that C(SUT) = 0 using the fact C(S) = C(T) = 0. I'm not really sure where to start this one though. First time I've seen anything like it and a quick search yielded no results about outer content at all.

I do have one theorem though. If S is a curve of finite length L, then C(S) = 0. I also figured ( not positive about this ) that C(∅) = 0.

 

[hedipaldi]

Attached

 

[STEMucator]

Attached

Hmm I sort of see what you're saying. I'm confused as to why you took your sums and unions up to ∞ rather than to n and then later argued as n → ∞, C(S) or C(T) → 0. So given any positive ε :

We take a set of rectangles R'_k such that T \subseteq \bigcup_{k=1}^{n} R_{k}^{'} and if I sum all the rectangles up to n, it will be smaller than (1/2)ε.

We take another set of rectangles R''_k such that S \subseteq \bigcup_{k=1}^{n} R_{k}^{''} and if we sum all these rectangles up to n it will also be smaller than (1/2)ε.

So hopefully I'm not mistaken here, but you asked me to consider the union of all the rectangles together.

So we take a set of rectangles R_k such that S \cup T \subseteq \bigcup_{k=1}^{n} {R_{k}^{'}} \cup {R_{k}^{''}} and if we sum all these rectangles, it will be less than (1/2)ε + (1/2)ε = ε.

 

[hedipaldi]

Yes this what i ment.