  We prove local background independence of the complete quantum closed string
field theory using the recursion relations for string vertices and the
existence of connections in CFT theory space. Indeed, with this data we
construct an antibracket preserving map between the state spaces of two nearby
conformal theories taking the corresponding string field measures $d\mu
e^{2S/\hbar}$ into each other. A geometrical construction of the map is
achieved by introducing a Batalin-Vilkovisky (BV) algebra on spaces of Riemann
surfaces, together with a map to the BV algebra of string functionals. The
conditions of background independence show that the field independent terms of
the master action arise from vacuum vertices $\V_{g,0}$, and that the overall
$\hbar$-independent normalization of the string field measure involves the
theory space connection. Our result puts on firm ground the widely believed
statement that string theories built from nearby conformal theories are
different states of the same theory.
