- #1
Redsummers
- 163
- 0
Homework Statement
Let [tex]\mathcal{A}[/tex] be σ-algebra over a set [tex]X[/tex], and μ a measure in [tex]\mathcal{A}[/tex].
Let [tex] A_{n} \in \mathcal{A}[/tex] with [tex]\sum_{n=1}^{\inf} \mu(A_{n})< \inf[/tex]
Show that this implies
μ ({[tex]x \in X[/tex] : [tex]x \in A_n[/tex] for infinitely many n}) = 0 .
The Attempt at a Solution
I don't even see how is the measure 0 if the measure of all A_n is a finite number...
I guess that the measure we want to show μ=0 is related with some kind of topology that makes it 0. What do they mean by
μ ({[tex]x \in X[/tex] : [tex]x \in A_n[/tex] for infinitely many n}) ?
"The measure of the points x in the set X s.t. x is an element of A_n (which the measure of A_n is finite) for infinitely many n."
I can't see how is this measure zero, so if I don't have a minimum intution I can't even attack the problem. Any suggestions?PS: which was the way to write LaTeX so that I can write something along a non-LaTeX text and still seem of the same size? i.e. how to make LaTeX text smaller.
Last edited: