## Fatou's Theorems

Let be Lebesgue Integrable and let

 (1)

be the corresponding Poisson Integral. Then Almost Everywhere in
 (2)

Let

 (3)

be regular for , and let the integral
 (4)

be bounded for . This condition is equivalent to the convergence of
 (5)

Then almost everywhere in ,
 (6)

Furthermore, is measurable, is Lebesgue Integrable, and the Fourier Series of is given by writing .

References

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 274, 1975.