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 for any help.
2007-11-20
04:18:16
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1 answers
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asked by
Sandra
1
in
Mathematics