on ${\mathbb{G}}$ with bounded support, we write $$|\nu| = \nu({\mathbb{G}}) \quad \text{and} \quad
\|\nu\|_\infty = \inf\{C>1; \; \|A\| < C \text{ for $\nu$-a.e.~$A\in{\mathbb{G}}$}\},$$ where $\|\mathord{\cdot}\|$ is some fixed operator norm.

\[p.existence\] Let $C>1$, $\delta>0$, and $\sigma$ be a $1$-rich measure with $\|\sigma \|_\infty \le C$. Then there exists $k \in {\mathbb{N}}$ with the following properties: If $\omega$ is a measure in ${\mathbb{G}}$ such that $|\omega| \le 1$ and $\|\omega\|_\infty \le C$ then $$\Lambda_k (\omega + \sigma) < |\omega| \delta + |\sigma| \log C.$$

The proposition implies that *given $A\colon (X,\mu) \to {\mathbb{G}}$, such that $\nu = A_* \mu$ is $1$-rich, there exists $T\in{\mathit{Aut}}(X,\mu)$ such that ${\mathit{LE}}(A,T)$ is small.* Indeed, take $\delta$ small. The measure $\sigma = \delta \nu$ is $1$-rich as well, so we can apply the proposition to $\omega = (1 - \delta) \nu$.

The fact that $k$ depends uniformly on $\omega$, provided $\|\omega\|_\infty \leq C$, will be important in the proof of theorem \[t.meas\].

For the reader’s convenience, we will give an informal sketch of the proof:

-   We first deal with the case where $\omega$ is a Dirac measure on some hyperbolic $H\in{\mathbb{G}}$. We use $1$-richness of $\sigma$ and the abstract lemma \[l.thouvenot\] to find products of 