Notation in subadditive ergodic theorem

Question

Let $\tau$ be a measure-preserving transformation on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and let $\{g_n\}$ be integrable with $$ g_{m+n}(\omega) \leq g_{m}(\omega)+g_{n}(\tau^m \omega) \quad \text{a.e.} \quad \omega \in \Omega $$ The textbook says that by defining $X_{m,n}=g_{n-m} \circ \tau^m$, the above inequality could be rewritten as $$ X_{0,n} \leq X_{0,m}+X_{m,n} $$ However, plugging $X$ into its definition directly seems to yield another result, i.e., $$ g_{n} \circ \tau^0 \leq g_{m} \circ \tau^0 +g_{n-m} \circ \tau^m $$ I am wondering how to understand this. Thanks!

Answer 1

It's a change of index. The initial inequality reads

$$g_{m+n} \leq g_m + g_n \circ \tau^m \quad \forall n, m \geq 0.$$

Now, let $m' := m$ and $n' := m+n$ I can replace $m$ with $m'$ and $n$ with $n'-m'$. This inequality becomes

$$g_{n'} \leq g_{m'} + g_{n'-m'} \circ \tau^{m'} \quad \forall n' \geq m' \geq 0.$$

Note that the quantification on $n$, $m$ changes. Finally, $\tau^0$ is the identity, so the later inequality is exactly the final inequality you got.