1
$\begingroup$

The entire problem reads:

(a) Find a sequence $f_n: [0,1] \rightarrow \mathbb{R}$ such that $\int_0 ^1 |f_n(x)| = 2$ for all $n \in \mathbb{N}$ and $\lim_{n \to \infty} f_n(x) = 1$ for all $x \in [0,1]$. (b) If the $f_n$ are as in part (a), then prove $\lim_{n \to \infty} \int_0 ^1 |f_n(x) -1| dx = 1$.

I'm having trouble finding the proper $f_n$ to fit this situation. Any help is appreciated. Once I find $f_n$, I may need tips for part (b) as well, but I think I could figure that part out given (a). Thanks in advance.

$\endgroup$
  • $\begingroup$ Try $f(x)=1+(n+2)(n+1)(1-x)x^n$. $\endgroup$ – Professor Vector Dec 11 '17 at 18:52
2
$\begingroup$

Define $f_n$ as a triangle on $[0,1/n]$ with $f(x) = 1$ for $x = 0$ and $x \in [1/n,1]$ (the triangle top is choosen such that the total integral is $2$).

$\endgroup$
1
$\begingroup$

For part b). By Egorov Theorem, for every $\epsilon>0$, there exists some measurable set $M_{\epsilon}\subseteq[0,1]$ such that $M_{\epsilon}^{c}:=[0,1]-M_{\epsilon}$ satisfies $|M_{\epsilon}^{c}|<\epsilon$ and $f_{n}\rightarrow 1$ uniformly on $M_{\epsilon}$.

We see that \begin{align*} \int_{0}^{1}|f_{n}-1|&=\int_{M_{\epsilon}}|f_{n}-1|+\int_{M_{\epsilon}^{c}}|f_{n}-1|\\ &\leq\int_{M_{\epsilon}}|f_{n}-1|+\int_{M_{\epsilon}^{c}}|f_{n}|+|M_{\epsilon}^{c}|\\ &=\int_{M_{\epsilon}}|f_{n}-1|+\int_{0}^{1}|f_{n}|-\int_{M_{\epsilon}}|f_{n}|+|M_{\epsilon}^{c}|\\ &=\int_{M_{\epsilon}}|f_{n}-1|-\int_{M_{\epsilon}}|f_{n}|+2+|M_{\epsilon}^{c}|\\ &<\int_{M_{\epsilon}}|f_{n}-1|-\int_{M_{\epsilon}}|f_{n}|+2+\epsilon, \end{align*} taking $n\rightarrow\infty$, we get \begin{align*} \limsup_{n}\int_{0}^{1}|f_{n}-1|\leq 0-1+2+\epsilon=1+\epsilon, \end{align*} so \begin{align*} \limsup_{n}\int_{0}^{1}|f_{n}-1|\leq 1. \end{align*} On the other hand, \begin{align*} \int_{0}^{1}|f_{n}-1|\geq\int_{0}^{1}|f_{n}|-1=2-1=1. \end{align*}

$\endgroup$
  • $\begingroup$ This makes total sense. I hadn't realized that it doesn't necessarily matters what $f_n$ is, only how it is defined in part (a). Thanks! $\endgroup$ – obewanjacobi Dec 11 '17 at 21:21
1
$\begingroup$

Hint for (b): Apply Fatou's lemma to $|f_n| + 1- |f_n-1|.$

$\endgroup$
0
$\begingroup$

Choose $$f(x):=4n^2x+1 \text{ for }x\in \left[0,\frac{1}{2n}\right]\\ f(x):=-4n^2x+1+4n \text{ for }x\in \left[\frac{1}{2n},\frac{1}{n}\right]\text{and }f(x)=1\text{ otherwise}.$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.