content_image/301923.jpg content_image/301922.jpg content_image/301921.jpg
Hence, 

<img src='content_image/301923.jpg'>

Since g$_{λ}$ ( x ; µ$_{0}$ ) ≤ 0 for every x ∈$_{R}$ , (63) implies that on $_{R}$\{ 0 } , g$_{λ}$ ( x ; µ$_{0}$ ) = 0 µ$_{0}$ -almost surely, so that g$_{λ}$ ( x ; µ$_{0}$ ) = 0 for all x ∈ S$_{µ}$$_{0}$ \{ 0 } follows immediately. 

The sufficiency part of Lemma 6 is established as follows. Suppose g$_{λ}$ ( x ; µ$_{0}$ ) ≤ 0 for every x ∈$_{R}$ . By integrating g$_{λ}$ ( x ; µ$_{0}$ ) w.r.t. µ$_{0}$ , we have 

<img src='content_image/301922.jpg'>

where (65) is due to (17) and g$_{λ}$ (0; µ$_{0}$ ) = f$_{λ}$ (0; µ$_{0}$ ) , and (66) follows from E$_{µ}$$_{0}$ $^{{}$X 2 } ≤ γ since µ$_{0}$ ∈ Λ( γ, q ) . Hence, λ E$_{µ}$$_{0}$ $^{{}$X 2 − γ } = 0 due to the fact that q < 1 . Furthermore, for any µ ∈ Λ( q ) , by integrating g$_{λ}$ ( x ; µ$_{0}$ ) w.r.t. µ , we have 

<img src='content_image/301921.jpg'>

Because g$_{λ}$ (0; µ$_{0}$ ) = f$_{λ}$ (0; µ$_{0}$ ) , we have E$_{µ}$ { f$_{λ}$ ( X ; µ$_{0}$ ) } ≤ 0 . Together with λ E$_{µ}$$_{0}$ $^{{}$X 2 − γ } = 0 and Lemma 5, this implies that µ$_{0}$ must be capacity-achieving. 

## C. Discreteness of µ$_{0}$ 

With Lemma 6 established, we now prove Property (c) in Theorem 1. 

Let λ ≥ 0 satisfy condition (58) and d ( z ; µ ) be defined in (23). We extend functions f$_{λ}$ ( x ; µ ) in Lemma 5 and g$_{λ}$ ( x ; µ ) in Lemma 6 to be defined on the whole complex plane C as (54) and (57), respectively, with x replaced by z ∈ $_{C}$. By Lemma 2, d ( z ; µ ) is a holomorphic function of z on $_{C}$, 

