Product Measures: miu and miu

[Smiling]

Homework Statement

We have two measures: miu and miu', with miu(E)=0 if E can be written as E' union E'' where the projection of E' onto the first coordinate is countable, and the projection of E'' onto the second coordinate is countable. If E cannot be written in this way, miu(E)=infinity.
Also, miu'(E)=0 if E can be written as E' union E'' union E''' where the projection of E' onto the first coordinate is countable, the projection of E'' onto the second coordinate is countable and the projection of E''' onto the line y=x is countable. If E cannot be written in this way, miu'(E)=infinity.

Now, let E={(x,-x)|x is in R}
Compute miu(E) and miu'(E)...

Homework Equations

The Attempt at a Solution

I need to show that miu(E)=infinity and that miu'(E)=0...

 

[lippyka]

For miu(E): We cannot write E as E' union E'' where the projection of E' onto the first coordinate is countable, and the projection of E'' onto the second coordinate is countable. Therefore, miu(E)=infinity. For miu'(E): E can be written as E' union E'' union E''' where the projection of E' onto the first coordinate is countable, the projection of E'' onto the second coordinate is countable and the projection of E''' onto the line y=x is countable. Thus, miu'(E)=0.