I am struggling with the very last part of theorem 5.4(pg.356) of TPE by Lehmann and Casella.
It is the theorem that proves the inadmissibility of the James Stein Estimator dominated by the positive part James Stein Estimator.
What I am struggling with is the last part of the proof of the theorem.
The followings are the TWO PARTS that I am struggling with.
I can't derive the last equality from the bottom third line of the proof.
I've carefully read the explanation(by S.Catterall Reinstate Monica) but I am still confused whether it is possible to derive the conditional expectation of continuous random variable by treating it as discrete.
Also I can't understand the bottom two lines of the proof.
I've found that it is an increasing function of the absolute value of |$\theta_1 y$|, but I can't understand how it completes the proof.