Is it true that if $ \Omega $ is a complete metric space, $F$ is its Borel $ \sigma $ -algebra, then for every positive $ \varepsilon $ there exists a compact $ K $ such that $ \mathbb{P}(K) > 1 - \varepsilon $?
First I thought that it is not true and tried to prove that using example with the space of ordinal numbers with a discrete metric:
Let $\Omega$ be the space of ordinal numbers with a discrete metric, from 0 to the first uncountable ordinal, $ 0, 1, 2, \ldots, \omega_1 $. Define the measure: $ \mathbb{P}(A) $ is equal to $ 1 $ or $ 0 $ depending on whether the set $ A $ contains all numbers $ \alpha $ greater than some $ \beta $, or not.
Then, the compact sets will be those containing only a finite number of elements, hence their measure is equal to $ 0 $. The statement is therefore false.
but then I saw that my measure is not countably additive
Maybe someone knows the other example where this statement is not true? Or how to prove that it's true? I think there is some example where it's not true
