I'm reading through "Real analysis for graduate students" by Bass. I am on the topic of outer measures and find myself confused. To ensure there isn't ambiguity in definitions/theorems I am going to state them here:
Proposition 4.2: suppose that $\mathcal{C}$ is a collection of subsets of $X$ such that $\emptyset\in\mathcal{C}$ and there exist $D_1,D_2,...\in\mathcal{C}$ such that $X=\bigcup_{i=1}^\infty D_i$. Suppose $\ell:\mathcal{C}\rightarrow [0,\infty]$ with $\ell(\emptyset)=0$. Define $$\mu^*(E)=\inf\{\sum_{i=1}^\infty \ell(A_i):A_i\in\mathcal{C}\ for\ each\ i \ and\ E\subset\bigcup A_i\}$$ Then $\mu^*$ is an outer measure.
Defintion 4.5: let $\mu^*$ be an outer measure. A set $A\subset X$ is $\mu^*$-measurable if $$\mu^*(E)=\mu^*(E\cap A)+\mu^*(E\cap A^c)$$for all $E\subset X$
Theorem 4.6: if $\mu^*$ is an outer measure on $X$, then the collection, $\mathcal{A}$ of $\mu^*$-measurable sets is a $\sigma$-algebra. Furthermore, if $\mu$ is the restriction of $\mu^*$ to $\mathcal{A}$ then $\mu$ is a measure.
My question is whether the following argument is correct: Let $X=\mathbb{N}$. let $\mathcal{C}=\{\emptyset, X\}$ let $\ell(A)$ be equal to the number of elements in $A$. Define $\mu^*$ as in proposition 4.2. This means that $\mu^*(A)=\infty$ for all subsets of $X$ with the exception of $\mu^*(\emptyset)$ which equals $0$. Applying definition 4.5 we see that all subsets of $X$ are $\mu^*$-measurable. (this is assuming, perhaps incorrectly, that $\infty=\infty+\infty$). Using theorem 4.6 we obtain the measure space $(X,\mathcal{P}(X),\mu)$ where $\mu$ is equal to $\infty$ for all subsets of $X$ with the exception of $\emptyset$ in which case $\mu(\emptyset)=0$
Is what I stated above accurate? Any help is greatly appreciated.
Yeah that's right. Or where do you have doubt?