Determinacy in L(R,μ)

Assume L(\mathbb{R},μ) satisfies ZF+DC+Θ>ω_2 + μis a normal fine measure on \powerset_{ω_1}(\mathbb{R}). The main result of this paper is the characterization theorem of L(\mathbb{R},μ) which states that L(\mathbb{R},μ) satisfies Θ>ω_2 if and only if L(\mathbb{R},μ) satisfies AD^+. As a result, we obtain the equiconsistency between the two theories: "ZFC + there are ω^2 Woodin cardinals" and "ZF+DC+μis a normal fine measure on \powerset_{ω_1}(\mathbb{R}) + Θ>ω_2".

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