... deLyra1
Email: delyra@latt.if.usp.br 
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... disk2
Post-publication note: it is important to observe here that, given any curve within the unit disk, from any internal point, say $z=0$, to $z_{1}$, the value of the integral does not depend on the curve. Given any two such curves, one can see this considering a small arc around $z_{1}$ connecting the two curves, using the Cauchy-Goursat theorem, and considering the limit in which the arc becomes infinitesimal. 
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... defined3
Post-publication note: it is important to observe here that the fact that the integral does not depend on the curve is implicitly being used here. 
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... number4
Post-publication note: it is important to observe here that we may also conclude that the resulting number does not depend on the curve, that is, on the direction along which the curve connects to $z_{1}$. 
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... have5
Post-publication note: it is important to observe that there is a subtlety here, because we are assuming for the identically zero real function $g(\theta)\equiv 0$ the result we are trying to prove for all integrable real functions. While it is immediately clear that these two identically zero functions do correspond to each other, the fact that the identically zero real function is the only real function associated to the identically zero inner analytic function is equivalent to the statement of the completeness of the Fourier basis. Therefore this proof must remain here subject to the fact that the Fourier basis is in fact complete. 
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... function6
Post-publication note: one may consider introducing here a further classification of the inner analytic functions. One could say that a regular inner analytic function is one that has, at all its singular points on the unit circle, the same status as the corresponding real function, regarding the fundamental analytic properties. By contrast, an irregular inner analytic function would be one that fails to have the same status as the corresponding real function, regarding one or more of the analytic properties, such as those of integrability, continuity, and differentiability. Important examples of irregular inner analytic functions would be those associated to singular distributions, as well as those associated to the examples of singular real functions which were mentioned in this section. 
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