Consider piecewise linear Lorenz maps on $[0, 1]$ of the following form \[
f_{a,b,c}(x)= {ll} ax+1-ac & x \in [0, c) b(x-c) & x \in (c, 1].\] We prove
that $f_{a,b,c}$ admits an absolutely continuous invariant probability measure
(acim) $\mu$ with respect to the Lebesgue measure if and only if $f_{a,b,c}(0)
\le f_{a,b,c}(1)$, i.e. $ac+(1-c)b \ge 1$. The acim is unique and ergodic
unless $f_{a,b,c}$ is conjugate to a rational rotation. The equivalence between
the acim and the Lebesgue measure is also fully investigated via the
renormalization theory.