5
votes
1answer
144 views

How large are measurable cardinals of higher orders?

For each ordinal $\alpha$ define the notions of $\alpha$ - measurable cardinals and $\alpha$ - normal measures as follows: A measure $\mu$ on a measurable cardinal $\kappa$ is a $0$-normal measure ...
3
votes
1answer
102 views

Existence of a probability measure under which a subset of $[0,1]$ cannot be approximated by a Borel set

Is there some probability measure, $p$, on $\left(\left[0,1\right],2^{\left[0,1\right]}\right)$ relative to which some $D\subseteq\left[0,1\right]$ cannot be approximated by a Borel set, i.e. if $E$ ...
3
votes
1answer
139 views

Measurable cardinal existence condition

We know that a cardinal $k > \omega$ is measurable if there is a measure function $\mu :2^k\mapsto \{0, 1\}$ that satisfies the following 3 conditions: 1.$<k$ -additive: for every set I of ...