# Comparing Measures on Finite & Countably Infinite Sets

• zeebo17
In summary, the conversation discusses the difference between two sets, \overline{F(A \times A)} and \overline{F(A) \times F(A)}, where A is a finite or countably infinite set. The bar notation represents the number of elements in the set. It is stated that for finite sets, the two sets have different cardinality, but for countably infinite sets, they are equal. The conversation also mentions the book's notation for F(A) as the power set and suggests proving that |F(A)|=2^|A|. It is also mentioned that the set structure does not matter and can be replaced with cardinal numbers and arithmetic operations.
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!

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

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

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.

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'."

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.

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.

zeebo17 said:
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.

## 1. What is the difference between finite and countably infinite sets?

Finite sets have a specific and limited number of elements, while countably infinite sets have an infinite number of elements that can be put into a one-to-one correspondence with the counting numbers (1, 2, 3, ...).

## 2. How do we compare measures on finite and countably infinite sets?

To compare measures on finite and countably infinite sets, we can use the concept of cardinality. Cardinality refers to the number of elements in a set, and the cardinality of a finite set will always be less than the cardinality of a countably infinite set.

## 3. Can we use the same methods to compare measures on both types of sets?

Yes, we can use the same methods to compare measures on both finite and countably infinite sets. However, since the cardinality of these sets is different, the methods may vary slightly in their application.

## 4. How do we measure the size of a countably infinite set?

The size of a countably infinite set is measured using cardinality. We can use the one-to-one correspondence method to compare the elements of a countably infinite set with the counting numbers and determine its cardinality.

## 5. Can there be a set that is both finite and countably infinite?

No, a set cannot be both finite and countably infinite. A set can only be categorized as either finite or countably infinite, as these terms represent two distinct and mutually exclusive types of sets.

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