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In this paper, we explore the structure theory of L(ℝ, μ) under the hypothesis L(ℝ, μ) ⊧ “AD + μ is a normal fine measure on ” and give some applications. First we show that “ ZFC + there exist ω2 Woodin cardinals”1 has the same consistency strength as “ AD + ω1 is ℝ-supercompact”. During this process we show that if L(ℝ, μ) ⊧ AD then in fact L(ℝ, μ) ⊧ AD+. Next we prove important properties of L(ℝ, μ) including Σ1 -reflection and the uniqueness of μ in L(ℝ, μ). Then we give the computation of full HOD in L(ℝ, μ). Finally, we use Σ1 -reflection and ℙmax forcing to construct a certain ideal on (or equivalently on in this situation) that has the same consistency strength as “ZFC+ there exist ω2 Woodin cardinals.”
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