I have an outer measure on an uncountable set $\Omega$ defined by $\mu^\ast(\varnothing) = 0$ and $\mu^\ast(A) = 1$ if $A \neq \varnothing$. I need to then determine the measurable sets under this outer-measure.
I'm not really sure what this means though I assume it's evident. I know that to determine the measurable sets I should be thinking Carathéodory's criterion; i.e let $T \subseteq \Omega$ and then $E$ is measurable if $$ \mu^\ast(T) \geq \mu^\ast(T \cap E) +\mu^\ast(T \cap E^c). $$ So here if $E$ is a measurable set then for every $T \subseteq \Omega$ I need the inequality to hold, can I argue that if $E \neq \varnothing$ then for every $T \neq \varnothing$ the inequality won't hold as you get $1 \geq 2$? But then what? Does this mean that the only measurable set under this outer measure is the empty-set? What's the point of such an outer measure? Thanks in advance for the clarification!