In this paper, we explore the structure theory ofL under the hypothesisL ⊧ “AD +μis a normal fine measure on” and give some applications. First we show that “ ZFC + there existω2Woodin cardinals”1has the same consistency strength as “ AD +ω1is ℝ-supercompact”. During this process we show that ifL ⊧ AD then in factL ⊧ AD+. Next we prove important properties ofL including Σ1-reflection and the uniqueness ofμinL. Then we give the computation of full HOD inL. Finally, we use (...) Σ1-reflection and ℙmaxforcing to construct a certain ideal on that has the same consistency strength as “ZFC+ there existω2Woodin cardinals.”. (shrink)