Let $ Ω = \mathbb{N} \times \mathbb{N} = { (i,j) : i,j ∈ \mathbb{N}}$,
and for any set $ A \subset Ω $ , let $μ(Α)$ be the number of distinct first coordinates among the points in $A$, i.e,
$μ(Α)$ = #( { $i:$ there exists $j ∈ \mathbb{N}$ such that $(i,j) ∈ A $} , $A \subset Ω$.
Note that $μ$ can be written as
$ μ(Α) =$ # $( π_1 (Α))$ $A \subset Ω$ ,
Where # is counting measure on $( \mathbb{N} , 2^ \mathbb{N} )$ and $π_1 : \mathbb{N} \times \mathbb{N} - > \mathbb{N}$ is the projection mapping each point in $\mathbb{N} \times \mathbb{N} $ onto it's first coordinate ,
i.e $π_1 ( ( i,j)) = i , (i,j) ∈ \mathbb{N} \times \mathbb{N}$.
Question I want to show the below:
i) $μ$ is an outer measure
ii) Characterize the class of μ-measurable sets.
and is μ a measure on $2^Ω$?