For f is an element of L-p(R-n) (1 <= p < infinity), the classical Littlewood-Paley g-function is defined by g(f)(x) = (integral(infinity)(0) vertical bar del u(x, t)vertical bar(2)t dt)(1/2), where u(x, t) denotes the Poisson integral of f. The following two weak type (1, 1) behaviors for the operator g are established: lambda m({x is an element of R-n : g(f)(x) > lambda}) less than or similar to n(3)parallel to f parallel to(1), lim(lambda -> 0+) lambda m({x is an element of R-n : g(f)(x) > lambda}) = root 2 c(n)omega(n-1)/2n vertical bar integral(Rn) f(x) dx vertical bar, for any f is an element of L-1(R-n), where c(n) is the constant in the Poisson kernel and omega(n-1) is the area of the unit sphere in R-n.