construction of stochastic integral

Question

Let $(\Omega,\mathcal F,P)$ be a probability space and $B_t$ be a Brownian motion on it. If a progressively measurable space $X_t$ satisfies $E\int_0^T|X_t|^2dt<\infty$, then the integral $\int_0^TX_tdB_t$ is a martingale. If we have $E(\int_0^T|X_t|^2dt)^{1/2}<\infty$, can we still construct the integral $\int_0^TX_tdB_t$ and is it still a martingale? Thanks!