  We study two questions. When does a function belong to the union of Lebesgue
spaces and when does a function have an $A_1$ majorant? We show these questions
are fundamentally related. For functions restricted to a fixed cube we prove
that the following are equivalent: a function belongs to $L^p$ for some $p>1$;
the function has an $A_1$ majorant; for any $p>1$ the function belongs to
$L^p_w$ for some $A_p$ weight $w$. We also examine the case of functions
defined on ${\mathbb R}^n$ and give characterizations of the union of $L^p_w$
over $w$ in $A_p$ and when a function has an $A_1$ majorant on all of ${\mathbb
R}^n$.
