Comparing Measures on Finite & Countably Infinite Sets

[zeebo17]

I just started learning some basic measure theory.

Could someone explain the difference between \overline{F(A \times A)} and \overline{F(A) \times F(A)} where A is a finite set. Also, how would this be different in A was an countably infinite set?

Thanks!

 

[rochfor1]

What is F(A)? This is not standard notation; you need to explain it before we can help.

 

[zeebo17]

I'm not sure, the book refers to F(A) as the power set.

 

[rochfor1]

Ok, then what's the bar over them? I figured there would be some sort of topological thing involved. Without the bars, say |A|=n. Then |A \times A|=n^2 and |F(A)|=2^n, so |F(A) \times F(A)|=(2^n)^2=4^n, but |F(A \times A)|=2^{ ( n^2 ) }, so these sets have different cardinality. If you let me know what the bar is, I can say more.

 

[zeebo17]

The bar means the number of elements in that set.I'm trying to understand what the difference is between \overline{F(A \times A)} and \overline{F(A) \times F(A)} so I can determine which has the most elements or which is "bigger'."

 

[rochfor1]

Ah well I used |A| for the number of elements of A. Note that |F(A)|=2^|A|. Try proving this; it shouldn't be very hard.

 

[zeebo17]

Ok, great- Thanks! I think I can get the rest from there.

The other thing I was wondering about was how to deal with that when A it is instead a countably infinite set. My book says that they would be equal in this case, but I'm not sure I see how.

 

[Hurkyl]

The other thing I was wondering about was how to deal with that when A it is instead a countably infinite set.

Again, it's just arithmetic of cardinal numbers. The set structure doesn't matter -- replace the sets with cardinal numbers, and the set arithmetic operations with the appropriate cardinal arithmetic operators.