Let x and y be two independent random variables. What is the difference between
(1) $P_x[\forall y, f(x,y) < \epsilon] >1- \delta$ (uniform bound), and
(2) $\forall y, P_x[f(x,y)<\epsilon]>1-\delta$ (not uniform bound)?
My understanding, and I'm looking for reassurance or correction:
(1) says that the total fraction of x-values that allows any of the $y$-values produce an $f(x,y)$ larger than $\epsilon$ is smaller than $\delta$. So we are talking about a single subset of $x$-values, and this has measure less than $\delta$.
(2) says that for any value of $y$ there is a fraction of at most $\delta$ of $x$-values that allows that particular $y$ to produce an $f(x,y)$ larger than $\epsilon$. So for each particular value of $y$ the set of size $\delta$ of $x$-values can be a different subset.
If this is so, and if my understanding is right, then it seems to me that for example,
A) Assuming that $E_y[f(x,y)|x]$ exists, Eq. (1) implies the following:
$P_x[E_y[f(x,y)] < \epsilon] >1- \delta$.
because I only need to discard a single set of x-values to have all the $f(x,y)$ below $\epsilon$, and hence to have its expectation wrt $y$ below $\epsilon$ too.
B) Assuming that $E_y[f(x,y) < \epsilon|x]$ exists,Eq. (1) also implies that:
$P_x[E_y[f(x,y) < \epsilon]] >1- \delta$.
Are these correct?