  We prove the existence of solutions for the stochastic differential equation
$dX_t=b(t,X_{t-})dZ_t+a(t,X_t)dt, X_0\in\R, t\ge 0,$ with only measurable
coefficients $a$ and $b$ satisfying the condition $0<\mu\le |b(t,x)|\le \nu$
and $|a(t,x)|\le K$ for all $t\ge 0, x\in\R$ where $\mu, \nu, $ and $K$ are
some constants. The driving process $Z$ is a symmetric stable process of index
$1<\alpha<2$. This generalizes the result of N. V. Krylov \cite{Krylov} for the
case of $\alpha=2$, that is when $Z$ is a Brownian motion. The proof is based
on integral estimates of Krylov type for the given equation which are also
derived in the note and are of independent interest. Moreover, unlike in
\cite{Krylov}, we use a different approach to derive the corresponding integral
estimates.
