 #1
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 36
Homework Statement:

Let ##X = \mathbb{R}##. For which ##\sigma##algebras is the following set function a measure:
##\nu(A) =
\begin{cases}
0 & \vert A \vert < \infty \\
1 & \vert A^c \vert < \infty \\
\end{cases}
##
Relevant Equations:

##nu## is a measure on ##\sigma##algebra ##\mathcal{A}## if
1) ##\nu(\emptyset) = 0## and
2) For any sequence of pairwise disjoint sets ##(A_n)_{n\in\mathbb{N}} \subset \mathcal{A}## we have ##\nu\left(\bigcup_{n\in\mathbb{N}} A_n \right) = \sum_{n\in\mathbb{N}} \nu(A_n)##.
Suppose ##\nu## is a measure on some ##\sigma##algebra ##\mathcal{A}##. Then we must have for all ##A \in \mathcal{A}## either ##A## or ##A^c## is finite, but not both. Because otherwise ##\nu(A)## is undefined or not well defined.
I've verified that ##\lbrace \emptyset, X \rbrace## and ##\lbrace \emptyset, X, A , A^c \rbrace## are suitable ##\sigma##algebras but i'm not sure how to rule out anything bigger? Can I please have a hint how to proceed?
For larger ##\sigma##algebras ##\mathcal{B}##, I consider when ##B, C \in \mathcal{B}## such that ##B, C## infinite , ##B^c, C^c## finite, and ##B, C## disjoint. Then
##\nu(B \cup C) = 1 \neq 1 + 1 = \nu(B) + \nu(C)##. But I can't think of a specific example of this happening?
I've verified that ##\lbrace \emptyset, X \rbrace## and ##\lbrace \emptyset, X, A , A^c \rbrace## are suitable ##\sigma##algebras but i'm not sure how to rule out anything bigger? Can I please have a hint how to proceed?
For larger ##\sigma##algebras ##\mathcal{B}##, I consider when ##B, C \in \mathcal{B}## such that ##B, C## infinite , ##B^c, C^c## finite, and ##B, C## disjoint. Then
##\nu(B \cup C) = 1 \neq 1 + 1 = \nu(B) + \nu(C)##. But I can't think of a specific example of this happening?