  Let $T$ be the Koopman operator of a measure preserving transformation
$\theta$ of a probability space $(X,\Sigma,\mu)$. We study the convergence
properties of the averages $M_nf:=\frac1n\sum_{k=0}^{n-1}T^kf$ when $f \in
L^r(\mu)$, $0<r<1$. We prove that if $\int |M_nf|^r d\mu \to 0$, then $f \in
\overline{(I-T)L^r}$, and show that the converse fails whenever $\theta$ is
ergodic aperiodic. When $\theta$ is invertible ergodic aperiodic, we show that
for $0<r<1$ there exists $f_r \in (I-T)L^r$ for which $M_nf_r$ does not
converge a.e. (although $\int |M_nf|^r d\mu \to 0$). We further establish that
for $1 \leq p <\frac{1}{r},$ there is a dense $G_\delta$ subset ${\mathcal
F}\subset L^p(X,\mu)$ such that $\limsup_n \frac{|T^nh|}{n^r}=\infty$ a.e. for
any $h \in {\mathcal F}$.
