  Assume L(\mathbb{R},\mu) satisfies ZF+DC+\Theta>\omega_2 + \mu is a normal
fine measure on \powerset_{\omega_1}(\mathbb{R}). The main result of this paper
is the characterization theorem of L(\mathbb{R},\mu) which states that
L(\mathbb{R},\mu) satisfies \Theta>\omega_2 if and only if L(\mathbb{R},\mu)
satisfies AD^+. As a result, we obtain the equiconsistency between the two
theories: "ZFC + there are \omega^2 Woodin cardinals" and "ZF+DC+\mu is a
normal fine measure on \powerset_{\omega_1}(\mathbb{R}) + \Theta>\omega_2".
