This thesis belongs to the field of descriptive inner model theory. Chapter 1 provides a

proper context for this thesis and gives a brief introduction to the theory of AD⁺, the theory

of hod mice, and a definition of K^J (R). In Chapter 2, we explore the theory of generalized

Solovay measures. We prove structure theorems concerning canonical models of the theory

AD⁺ + there is a generalized Solovay measure" and compute the exact consistency strength

of this theory. We also give some applications relating generalized Solovay measures to the

determinacy of a class of long games. In Chapter 3, we give a HOD analysis of AD⁺ + V =

L(P(R)) models below AD_R + Theta is regular." This is an application of the theory of hod

mice developed in G. Sargsyan's thesis. We also analyze HOD of AD⁺-models of the form V = L(R; mu)

where mu is a generalized Solovay measure. In Chapter 4, we develop techniques for the core

model induction. We use this to prove a characterization of AD⁺ in models of the form

V = L(R;mu), where mu is a generalized Solovay measure. Using this framework, we also can

construct models of AD_R + Theta is regular" from the theory ZF + DC + Theta is regular + omega₁ is

P(R)-supercompact". In fact, we succeed in going further, namely we can construct a model

of AD_R + Theta is measurable" and show that this is in fact, an equiconsistency.