User:Kcbetancourt/AnalysisHW5

4.3 Let $$ f $$ be a nonnegative measurable function. Show that $$ \int f= 0 $$ implies that $$ f= 0 $$ a.e.

4.5 Let $$ f $$ be a nonnegative integrable function. Show that the function $$ F $$ defined by $$ F(x) = \int_{-\infty}^{x} f $$ is continuous by using Theorem 10.

(Theorem 10 - Monotone Convergence Theorem: Let $$ \left\{{f_n}\right\} $$ be an increasing sequence of nonnegative measurable functions, and let $$ f = lim f_n $$ a.e. Then $$ \int f = lim\int f_n $$.)

4.8 Prove the following generalization of Fatou's Lemma: If $$ \left\{{f_n}\right\} $$ is a sequence of nonnegative functions, then \int lim inf f_n \le lim inf \int f_n.