Let (X, M u) be a measure space, where X is a set, M is a sigma-algebra on X and u is a measure defined on M. Let f_n be a sequence of functions defined on X and with values on [0, oo] such that lim f_n = f. Suppose that lim Integral f_n du = Integral f du < oo (the integrals taken over X).
Show that, for every set E of M, Integral_E f_n du = Integral_E f du, where Integral_E means integral over E. Also, show that this conclusion may fail if we have lim Integral f_n du = Integral f du = oo
Thank you.
2007-12-20
04:03:23
·
2 answers
·
asked by
Laura
1