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 $$ \int \lim \inf f_n \le \lim \inf \int f_n $$.

4.14

a.) Show that under the hypotheses of Theorem 17 we have $$ \int \left|{f_n-f}\right| \to 0 $$.

b.) Let $$ \left\{{f_n}\right\} $$ be a sequence of integrable functions such that $$ f_n \to f $$ a.e. with $$ f $$ integrable. Then $$ \int \left|{f-f_n}\right| \to 0 \iff \int \left|{f_n}\right| \to \int \left|{f}\right| $$.

4.15

a.) Let $$ f $$ be integrable over $$ E $$. Then, given $$ \varepsilon > 0 $$, there is a simple function $$ \phi $$ such that $$ \int_E \left|{f-\phi }\right| < \varepsilon $$. [Apply Problem 4 to the positive and negative parts of $$ f $$.]

b.) Under the same hypothesis there is a step function $$ \psi $$ such that $$ \int_E \left|{f-\psi }\right| < \varepsilon $$. [Combine part (a) with Proposition 3.22.]

c.) Under the same hypothesis there is a continuous function $$ g $$ vanishing outside a finite interval such that $$ \int_E \left|{f-g}\right| < \varepsilon $$.

16. Establish the Riemann-Lebesgue Theorem: If $$ f $$ is an integrable function on $$ (-\infty, \infty ) $$, then $$ \lim_{n\to \infty} \int_{-\infty}^\infty f(x) \cos nx dx = 0 $$. [Hint: The theorem is easy if $$ f $$ is a step function. Use Problem 15.]

Also, Find an example of a sequence $$ \left\{{f_n}\right\} $$ of bounded functions on [0,1] that tends to 0 in measure, but does not converge anywhere, i.e. find a sequence of functions on [0,1] that converges in measure but does not converge pointwise anywhere.