  Given a Radon measure $\mu$ on $R^d$, which may be non doubling, we introduce
a space of type BMO with respect to this measure. It is shown that many
properties that hold when $\mu$ is doubling remain valid for the space BMO
introduced in this paper, without assuming $\mu$ doubling. For instance,
Calderon-Zygmund operators which are bounded in $L^2$ are bounded from
$L^\infty$ into the new BMO space. Moreover, a John-Nirenberg inequality is
satisfied, and the predual of BMO is an atomic space $H^1$. Using a sharp
maximal function it is proved that operators bounded from $L^\infty$ into BMO
and from $H^1$ into $L^1$ are also bounded on $L^p$, $1<p<\infty$. This result
gives a new proof of the T(1) theorem for the Cauchy transform with non
doubling measures. Finally, a result about commutators is obtained.
