Show that a distribution function is dominated

Solution 1:

The measure you are looking for is the counting measure on the integers (or on the support of the measure associated to $F$). Notice that since $F_p$ takes all its mass on $\mathbb{Z}_{\geq 1}$, if the counting measure of a set is zero, then so is its probability under the measure associated to $F_p$.

Proving series converges using Fourier series

Finding the covariance matrix of a multivariate gaussian

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If $f^{-1}((r,\infty))$ is measurable for all $r$ in $\mathbb{Q}$, prove that $f$ is measurable

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Let $Y_n,X_n$ be a sequence of r.v. Does $\sup_z \Big|P(Y_n<z\ |\ X_n) - \Phi(z)\Big|\rightarrow_p0 \implies \{Y_n|X_n=x_n\}\xrightarrow[]{d}N(0,1)?$