content_image/319414.jpg content_image/319413.jpg content_image/319415.jpg content_image/319416.jpg content_image/319412.jpg
Combining (29) and (33) yields that 

<img src='content_image/319414.jpg'>

which is integrable. Therefore, the integral ∫ ∞ −∞ φ ( y − z ) log p$_{Y}$ ( y ; µ )d y is uniformly convergent for all z ∈ U . Moreover, φ ( y − z ) log p$_{Y}$ ( y ; µ ) is a holomorphic function of z on U for each y ∈$_{R}$ . According to the differentiation lemma [18], ξ ( z ) is a holomorphic function of z on U . It then follows that it is holomorphic on the whole complex plane$_{C}$. Lemma 2 is thus established. 

Let F ( µ ) be a real-valued function defined on the convex set Λ( q ) and µ$_{0}$ ∈ Λ( q ) . Define the weak derivative of F ( µ ) at µ$_{0}$ as 

<img src='content_image/319413.jpg'>

whenever the limit exists. The following result, which finds its parallel in [6], [9], [10] gives the weak derivative of the mutual information function I ( µ ) . 

Lemma 3: Let µ$_{0}$, µ ∈ Λ( q ) , the weak derivative of the mutual information function I ( µ ) at µ$_{0}$ is 

Proof: Define µ$_{θ}$ = (1 − θ ) µ$_{0}$ + θµ for all θ ∈ (0 , 1] . It can be shown that 

<img src='content_image/319415.jpg'>

<img src='content_image/319416.jpg'>

Therefore, it suffices to show that 

<img src='content_image/319412.jpg'>

In the remainder of this proof, we find a function independent of θ that dominates the integrand so that dominated convergence theorem can be used to establish (39) by exchanging the order of the limit and the integral therein. 

