I am reading Royden textbook. He defines a measure to be a set function $\mu: \Sigma \to [0, \infty]$ on a measurable space $(X, \Sigma, \mu)$, such that $\mu(\emptyset) = 0$ and countable additivity holds.
In chatper 17.3 – 5, a set function $\mu: S \to [0, \infty]$ can be extended to caratheodory measure $\bar{\mu}$ on the conditions that $S$ is a semiring, and $\mu$ have countable monotonicity and finite additivity properties.
$\bar{\mu}$ is defined to be an outer measure $\mu^*$ restricted to $\Sigma$.
My question is that is caratheodory measure $\bar{\mu}$ is the same as a measure ? If yes, how can we show the countable additivity property of $\bar{\mu}$?
Best Answer
Caratheodory measure $ \bar{\mu}$ is not same as original measure $\mu$ as both measures are defined on different collection of subsets ($\mu$ is defined on semiring $S$ and $\bar\mu$ is defined on $\sigma$-algebra $\Sigma$).
But note that by construction $ S \subset \Sigma$. Hence what we can say is $\bar\mu$ is same as $\mu$ when $\bar\mu$ is restricted to $S$. By using Caratheodory condition you can prove that $\bar\mu$ is countable additive on $\Sigma$.