Theorem Let E be a finite measure Suppose hn is a sequence o

Theorem: Let E be a finite measure. Suppose {h_n} is a sequence of nonnegative integrable functions that converges pointwise a.e. on E to h 0. Then lim_n righarrow infinity integral_E h_n = 0 if and only if {h_n} is uniformly integrable over E. Show that the Theorem is false without the assumption that the h_n is nonnegative.

Solution

If we assume that the the sequence {hn} is not non-negative, then lim integral (hn) not becomes zero, thus the theorem does not holds.