  Let $\mu$ be a Borel measure on $R^d$ which may be non doubling. The only
condition that $\mu$ must satisfy is $\mu(B(x,r))\leq C r^n$ for all $x\in
R^d$, $r>0$, and for some fixed $0<n\leq d$. In this paper, we develop
Littlewood-Paley theory for functions in $L^p(\mu)$. One of the main
difficulties is the construction of reasonable approximations of the identity
for obtaining a Calderon type reproducing formula. Moreover, it is shown that
the T(1) theorem for n-dimensional Calderon-Zygmund operators, without doubling
assumptions, can be proved using the Littlewood-Paley decomposition that is
obtained for $L^2(\mu)$ functions, as in the classical case of homogeneous
spaces.
