I need to give an example of a measure &#x03BC;<!-- μ --> and subsets A n </

generiranfi7qu

generiranfi7qu

Answered question

2022-05-11

I need to give an example of a measure μ and subsets A n s.t.
A 1 A 2 . . .
and
μ ( n = 1 A n ) lim n μ ( A n )
I hat the following in mind. Take μ to be the counting measure and A 1 = N , A 2 = N { 1 } , . . . then
μ ( n = 1 A n ) = μ ( ) = 0
but now I have some strugles to show the other part with the limes. could someone help me please? It would be nice if we could procede with this example since this was my own idea without using any internet.

Answer & Explanation

Gia Schaefer

Gia Schaefer

Beginner2022-05-12Added 16 answers

Well, since A n = N { 1 , , n } has infinitely many elements for any n, we must have
μ ( A n ) = .
Now what can you conclude about lim n μ ( A n )?

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