Consider the following sets: $$ A = \Big\{ x\in X: \Pr\bigg(\lim_{n \to \infty}d\big(p_n, [\ell(x), u(x) ] \big)= 0\bigg)=1 \Big\}, $$ and $$ A_n = \Big\{ x\in X: d\big(p_n, [\ell(x), u(x) ] \big)= 0\Big\}, $$ where:

• $\Pr$ denotes probability.
• $X\subseteq \mathbb{R}$.
• $(p_n)_n$ is a sequence of random variables taking values in $[0,1]$.
• $\ell(\cdot)$ and $u(\cdot)$ are real function taking values in $[0,1]$.
• $d\big(p_n, [\ell(x), u(x) ] \big):= \inf \big\{|p_n - y| : y \in [\ell(x), u(x) ] \big\}$.

Assumptions:

Ass1. $A$ is non-empty.

Ass2. $\forall x\in A$ and $\delta>0$, $\Pr(x\in A_n(\delta))\geq 1-\frac{2}{\exp(2n \delta^2)}$, where $$ A_n(\delta) = \Big\{ x\in X: d\big(p_n, [\ell(x)-\delta, u(x)+\delta ] \big)= 0\Big\}. $$

Could you help me to show that under Ass1 and Ass2 $$ d_H(A, A_n)\rightarrow_{a.s.} 0, $$ where $$ d_H(A, A_n)\equiv \max\{\sup_{x\in A_n}d(x,A), \sup_{x\in A}d(x, A_n)\}, $$ is the Hausdorff distance.

Consider the following sets: $$ A = \Big\{ x\in X: \Pr\bigg(\lim_{n \to \infty}d\big(p_n, [\ell(x), u(x) ] \big)= 0\bigg)=1 \Big\}, $$ and $$ A_n = \Big\{ x\in X: d\big(p_n, [\ell(x), u(x) ] \big)= 0\Big\}, $$ where:

• $\Pr$ denotes probability.
• $X\subseteq \mathbb{R}$.
• $(p_n)_n$ is a sequence of random variables taking values in $[0,1]$.
• $\ell(\cdot)$ and $u(\cdot)$ are real function taking values in $[0,1]$.
• $d\big(p_n, [\ell(x), u(x) ] \big):= \inf \big\{|p_n - y| : y \in [\ell(x), u(x) ] \big\}$.

Could you help me to show that $$ d_H(A, A_n)\rightarrow_{a.s.} 0, $$ where $$ d_H(A, A_n)\equiv \max\{\sup_{x\in A_n}d(x,A), \sup_{x\in A}d(x, A_n)\}, $$ is the Hausdorff distance.

Consider the following sets: $$ A = \Big\{ x\in X: \Pr\bigg(\lim_{n \to \infty}d\big(p_n, [\ell(x), u(x) ] \big)= 0\bigg)=1 \Big\}, $$ and $$ A_n = \Big\{ x\in X: d\big(p_n, [\ell(x), u(x) ] \big)= 0\Big\}, $$ where:

• $\Pr$ denotes probability.
• $X\subseteq \mathbb{R}$.
• $(p_n)_n$ is a sequence of random variables taking values in $[0,1]$.
• $\ell(\cdot)$ and $u(\cdot)$ are real function taking values in $[0,1]$.
• $d\big(p_n, [\ell(x), u(x) ] \big):= \inf \big\{|p_n - y| : y \in [\ell(x), u(x) ] \big\}$.
• A is non-empty.

Could you help me to show that $$ d_H(A, A_n)\rightarrow_{a.s.} 0, $$ where $$ d_H(A, A_n)\equiv \max\{\sup_{x\in A_n}d(x,A), \sup_{x\in A}d(x, A_n)\}, $$ is the Hausdorff distance.

Additional details which may be helpful:

• Note that $A$ is assumed non-empty.

• Also note that $(p_n)_n$ does not necessarily converge.

• Lastly, I have shown that, $\forall x\in A$ and $\delta>0$, $\Pr(x\in A_n(\delta))\geq 1-\frac{2}{\exp(2n \delta^2)}$, where $$ A_n(\delta) = \Big\{ x\in X: d\big(p_n, [\ell(x)-\delta, u(x)+\delta ] \big)= 0\Big\}. $$

Consider the following sets: $$ A = \Big\{ x\in X: \Pr\bigg(\lim_{n \to \infty}d\big(p_n, [\ell(x), u(x) ] \big)= 0\bigg)=1 \Big\}, $$ and $$ A_n = \Big\{ x\in X: d\big(p_n, [\ell(x), u(x) ] \big)= 0\Big\}, $$ where:

• $\Pr$ denotes probability.
• $X\subseteq \mathbb{R}$.
• $(p_n)_n$ is a sequence of random variables taking values in $[0,1]$.
• $\ell(\cdot)$ and $u(\cdot)$ are real function taking values in $[0,1]$.
• $d\big(p_n, [\ell(x), u(x) ] \big):= \inf \big\{|p_n - y| : y \in [\ell(x), u(x) ] \big\}$.

Assumptions:

Ass1. $A$ is non-empty.

Ass2. $\forall x\in A$ and $\delta>0$, $\Pr(x\in A_n(\delta))\geq 1-\frac{2}{\exp(2n \delta^2)}$, where $$ A_n(\delta) = \Big\{ x\in X: d\big(p_n, [\ell(x)-\delta, u(x)+\delta ] \big)= 0\Big\}. $$

Could you help me to show that under Ass1 and Ass2 $$ d_H(A, A_n)\rightarrow_{a.s.} 0, $$ where $$ d_H(A, A_n)\equiv \max\{\sup_{x\in A_n}d(x,A), \sup_{x\in A}d(x, A_n)\}, $$ is the Hausdorff distance.