Ergodic averages are measurable, representative independent, and Lp contractive

Measurability of $T$ makes $f\circ T$ measurable. If $f=g$ off the measurable null set $N$, then $f\circ T=g\circ T$ off $T^{-1}N$, and $\mu (T^{-1}N)=\mu (N)=0$. Thus $U_T$ is representative independent. Pointwise composition distributes over addition and scalar multiplication, so the descended map is linear over either scalar field.

For $1\le p<\mathrm{\infty }$, integral invariance applied to the nonnegative measurable function $\mathrm{\mid }f{\mathrm{\mid }}^p$ gives $\parallel f\circ T{\parallel }_p^p=\int \mathrm{\mid }f{\mathrm{\mid }}^p\circ Td\mu =\int \mathrm{\mid }f{\mathrm{\mid }}^{p}d\mu =\parallel f{\parallel }_p^p\mathrm{.}$ Taking the nonnegative $p$th root proves equality of the norms.

For $p=\mathrm{\infty }$ and every finite $M\ge 0$, the exceptional set for $\mathrm{\mid }f\circ T\mathrm{\mid }\le M$ is $T^{-1}\{\mathrm{\mid }f\mathrm{\mid }>M\}$. Its measure equals that of $\{\mathrm{\mid }f\mathrm{\mid }>M\}$. Hence $M$ is an essential bound for $f\circ T$ exactly when it is one for $f$, and their infima are equal.

Iterating step 1.1 shows that every $U_T^k[f]=[f\circ T^k]$ is well defined and has norm $\parallel f{\parallel }_p$. Finite linear combinations therefore make $S_{n}f$ and $A_{n}f$ well-defined measurable classes.

The real or complex Minkowski inequality and positive homogeneity now give $\parallel A_{n}f{\parallel }_p\le \frac{1}{n}{\sum }_{k=0}^{n-1}\parallel U_T^{k}f{\parallel }_p=\parallel f{\parallel }_p\mathrm{.}$ This includes $n=1$ and $p=\mathrm{\infty }$, and no inverse of $T$ was used.