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We study the boundary behavior of functions in the Hardy spaces on theinfinite dimensional polydisk. These spaces are intimately related to the Hardyspaces of Dirichlet series. We exhibit several Fatou and Marcinkiewicz-Zygmundtype theorems for radial convergence. As a consequence one obtains easy newproofs of the brothers F. and M. Riesz theorems in infinite dimension. Finally,we provide counterexamples showing that the pointwise Fatou theorem is not truein infinite dimensions without restrictions to the mode of radial convergenceeven for bounded analytic functions.

Fatou and brother Riesz theorems in the infinite-dimensional polydisc