Denote by $\Phi(x)$ the cumulative distribution function of $\Gamma$, the standard normal distribution. More explicitly, $$ \Phi(x)=\int_{-\infty}^x \Gamma[dt]:=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-t^2/2} dt. $$

It defines a homeomorphisms $\Phi:\bR\to (0,1)$. Its inverse $Q=\Phi^{-1}$ is the so called quantile function of $\Gamma$. If $\mu$ is the uniform distribution on $[0,1]$, then $Q_\#\mu$, the push forward of $\mu$ is the normal distribution. The push forward is defined as in Iosif Pinelis' answer, $\newcommand{\bR}{\mathbb{R}}$

$$Q_\#\mu[S]= \mu\big[ Q^{-1}(S)\big],$$

for any Borel subset $S\subset \bR$.

Take an ergodic transformation of $T$ of the $\big((0,1),\mu\big)$. Then $Q\circ T \circ \Phi$ is an ergodic transformation of $(\bR,\Gamma)$.

Denote by $\Phi(x)$ the cumulative distribution function of $\Gamma$, the standard normal distribution. More explicitly, $$ \Phi(x)=\int_{-\infty}^x \Gamma[dt]:=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-t^2/2} dt. $$

$\newcommand{\bR}{\mathbb{R}}$ It defines a homeomorphisms $\Phi:\bR\to (0,1)$. Its inverse $Q=\Phi^{-1}$ is the so called quantile function of $\Gamma$. If $\mu$ is the uniform distribution on $[0,1]$, then $Q_\#\mu$, the push forward of $\mu$ is the normal distribution. The push forward is defined as in Iosif Pinelis' answer,

$$Q_\#\mu[S]= \mu\big[ Q^{-1}(S)\big],$$

for any Borel subset $S\subset \bR$.

Take an ergodic transformation of $T$ of the $\big((0,1),\mu\big)$. Then $Q\circ T \circ \Phi$ is an ergodic transformation of $(\bR,\Gamma)$.