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 .