Reflection Invariance of Lebesgue Measure and Symmetry of the Standard Normal Distribution

For a subset $A$ of the real line $\mathbb{R}$ write $-A=\{-x:x\in A\}$. Let $\lambda^{*}$ be the Lebesgue outer measure, $\lambda$ Lebesgue measure, $\mathcal{B}(\mathbb{R})$ the Borel $\sigma$-algebra, and $N$ the standard normal distribution. Then:

1. (Reflection invariance of Lebesgue measure) For every $A\subseteq\mathbb{R}$, $\lambda^{*}(-A)=\lambda^{*}(A)$. For every Borel set $B$, the reflected set $-B$ is Borel and $\lambda(-B)=\lambda(B)$.

2. (Reflection invariance of the standard normal distribution) For every Borel set $B$,

3. (Symmetry) If $Z$ is a standard normal random variable on a probability space, then $-Z$ is a standard normal random variable.