Is there a probability measure on a set whose cardinality is $2^{\mathfrak{c}}$ ?
I considered a real-valued stochastic process $S(t),t\in[0,1]$. If $t_0$ is a constant, $S(t_0)$ is an ordinary random variable.
But suppose the set of all sample curves of $S(t)$ is $SC$. We have $Card(SC)=2^{\mathfrak{c}}$.
Let $SP = \{f(t_0)| f\in SC\}$.
I think the probability distribution of random variable $S(t_0)$ should be consistent with the measure of different values in the set $SP$. But $Card(SP)=2^{\mathfrak{c}}$ and the probability measure I know is defined on the set of real numbers.
How to understand the meaning of the set $SP$ and the relationship between $SP$ and $S(t_0)$?