On an identity for stochastic integrals

A sufficient condition is obtained for the identity
$$ \mathbf{M}\exp\biggl\{\int_0^T f(t,\omega)\,dw(t)-\frac12\int_0^T f^2(t,\omega)\,dt\biggr\}=1 $$
to hold. (Here $\int_0^T f(t,\omega)\,dw(t)$ is the stochastic integral with respect to a Wiener process $w(t)$.) This condition is shown to be close to a necessary one.