We recall that $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$ is a measure, and that for $A \in \operatorname{Int}$, $\mu(A) = \int \mathbb{1}_A(x) \mu(x) dx$. Let $A \in \operatorname{Int}$. For $t \geqslant 0$, we define the set $A _ { t } = \{ x \in \mathbb { R } : d ( x , A ) \leqslant t \}$. 16a. Show that $A _ { t } \in \operatorname { Int }$ for all $t \geqslant 0$. 16b. We further assume that $\mu ( A ) > 0$. Show that for all $t \geqslant 0$, we have $$1 - \mu \left( A _ { t } \right) \leqslant \frac { e ^ { - t ^ { 2 } / 2 } } { \mu ( A ) }$$
We recall that $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$ is a measure, and that for $A \in \operatorname{Int}$, $\mu(A) = \int \mathbb{1}_A(x) \mu(x) dx$. Let $A \in \operatorname{Int}$. For $t \geqslant 0$, we define the set $A _ { t } = \{ x \in \mathbb { R } : d ( x , A ) \leqslant t \}$.
16a. Show that $A _ { t } \in \operatorname { Int }$ for all $t \geqslant 0$.
16b. We further assume that $\mu ( A ) > 0$. Show that for all $t \geqslant 0$, we have
$$1 - \mu \left( A _ { t } \right) \leqslant \frac { e ^ { - t ^ { 2 } / 2 } } { \mu ( A ) }$$