  Given a finite measure space $(\Omega,\Sigma,\mu)$, we show that any Banach
space $X(\mu)$ consisting of (equivalence classes of) real measurable functions
defined on $\Omega$ such that $f \chi_A \in X(\mu) $ and $ \|f \chi_A \| \leq
\|f\|, \, f \in X(\mu), \ A \in \Sigma$, and having the subsequence property,
is in fact an ideal of measurable functions and has an equivalent norm under
which it is a Banach function space. As an application we characterize norms
that are equivalent to a Banach function space norm.
