# Homogenization of a quasimorphism

Suppose $f:G \to \R$ is a quasimorphism. The homogenization of $f$ is a homogeneous quasimorphism $\mu_f$ (sometimes denoted $\overline{f}$) defined as follows:
$\mu_f := x \mapsto \lim_{n \to \infty} \frac{f(x^n)}{n}$
• Upper bound: The defect $D(\mu_f)$ of the homogenization is at most twice the defect $D(f)$ of $f$.