measure theory problems and step functions

Question

I have several questions that I haven't worked out. Any hints or solutions will be appreciated.

1. Suppose that {$f_n$} is a sequence of real valued continuously differentiable functions on [$0,1$] such that $ \|f_n\|_1 \to 0$ as $n\to\infty$ and $ \|f'_n\|_1 \to 0$ as $n\to\infty$, i.e. both $f_n$ and the derivatives converge in $L_1$ norm. Prove that $f_n$ converges uniformly to zero on [$0,1$].

I'm not sure how to start in this one. And for the next problem,

2. Investigate the convergence of $\Sigma a_n$, where $a_n =\int_{0}^{1} \frac{x^n sin(\pi x)}{1-x} dx$.

I'm thinking to change $\frac{1}{1-x}$ to $\Sigma x^k$ and then exchange the integral and the sum but it came back to the original form. Or I just take the sum of $a_n$ switch the sum and integral since the integrand is nonnegative and measurable function. Then the integrand becomes $\frac{sin(\pi x)}{(1-x)^2}$,so I think the series should diverge.

3. Let $f \in L^1 (m)$. For $k=1,2,3,...$, let $f_k$ be the step function defined by $f_k (x) = k\int_{j/k}^{{j+1}/k} f(t)dt$ for $\frac{j}{k}<x<\frac{j+1}{k}$, $j=0,\pm1,...$. Show that $f_k$ converges to $f$ in $L_1$ norm.

This one seems more direct but when I do the $\|f_n - f\|_1$, I have trouble switching the two integrals.

Answer 1

One question per post.

1. For $0\le x\le y\le1$ $$ |f_n(x)-f_n(y)|=\Bigl|\int_x^yf_n'(t)\,dt\Bigr|\le\|f'_n\|_1. $$ Thus $\{f_n\}$ is equicontinuous and has a uniformly convergent subsequence. Since $\{f_n\}$ converges to $0$ in $L^1$, the uniform limit of the aforementioned subsequence must be $0$. This argument can be carried out for any subsequence of $\{f_n\}$, that is, we can prove that any subsequence of $\{f_n\}$ has a subsequence that converges uniformly to $0$. This imples that the whole sequence $\{f_n\}$ converges uniformly to $0$.

Answer 2

Let $(f_{n_k})$ be a subsequence of $(f_n)$. Pick a subsequence of $(f_{n_k})$, call it $(g_n)$, that converges a.e. to $0$ (this can be done since $(f_n)$ converges to $0$ in $L_1$).

Pick $y\in[0,1]$ with $\lim\limits_{n\rightarrow\infty} g_n(y)=0$. Then for any $n$ and any $x\in[0,1]$ $$ |g_n(x)-g_n(y)|\le\biggl|\,\int_y^x g_n'(x)\,\biggr| \le\Vert g_n'\Vert_{L_1}. $$

The right hand side of the above can be made as small as you wish by taking $n$ sufficiently large. Since $g_n(y)\rightarrow 0$, this implies $|g_n(x)|$ has the same property; i.e., $(g_n)$ converges uniformly to $0$.

So, every subsequence of $(f_n)$ has a subsequence that converges uniformly to $0$. Thus $(f_n)$ converges uniformly to $0$.