consequence of this example, one sees that for 0 < p < 1, one cannot characterize the real-Hp(T) (defined below) in the simple way given above, but must use the actual definition using maximal functions, which is given further along somewhere below. For the same function F, let fr(eiθ) = F(reiθ). The limit when r → 1 of Re(fr), in the sense of distributions on the circle, is a non-zero multiple of the Dirac distribution at z = 1. The Dirac distribution at a point of the unit circle belongs to real-Hp(T) for every p < 1 (see below). Factorization into inner and outer functions (Beurling) For 0 < p ≤ ∞, every non-zero function f in Hp can be written as the product f = Gh where G is an outer function and h is an inner function, as defined below . This "Beurling factorization" allows the Hardy space to be completely characterized by the spaces of inner and outer functions. One says that G(z) is an outer (exterior) function if it takes the form for some complex number c with |c| = 1, and some positive measurable function on the unit circle such that is integrable on the circle. In particular, when is integrable on the circle, G is in H1 because the above takes the form of the Poisson kernel . This implies that for almost every θ. One says that h is an inner (interior) function if and only if |h| ≤ 1 on the unit disk and the limit exists for almost all θ and its modulus is equal to 1 a.e. In particular, h is in H∞. The inner function can be further factored into a form involving a Blaschke product. The function f, decomposed as f = Gh, is in Hp if and only if φ belongs to Lp(T), where φ is the positive function in the representation of the outer function G. Let G be an outer function represented as above from a function φ on the circle. Replacing φ by φα, α > 0, a family (Gα) of outer functions is obtained, with the properties: G1 = G, Gα+β = Gα Gβ and |Gα| = |G|α almost everywhere on the circle. It follows that whenever 0 < p, q, r < ∞ and 1/r = 1/p + 1/q, every function f in Hr can be expressed as the product of a