Show that: $ \int_{[a,b]} \liminf_{n\rightarrow \infty} f_n(x) d\lambda(x)\le \lim_{\varepsilon\searrow0}(f(b-\varepsilon)-f(a+\varepsilon)) $

Question

Let $\lambda$ be Lebesgue measure on $\mathbb{R}$. $f$: [$a$,$b$] $\rightarrow$ $\mathbb{R}$ measurable, $\lambda$-almost everywhere differentiable and monotonic increasing.

We define: $\bar{f}$ : $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$ :

$$\bar{f}(x) = \begin{cases} \lim_{x\searrow a} f(x) , &\ x \le a\\ \ \\ f(x),&\ x \in (a,b) \\ \ \\ \lim_{x\nearrow b} f(x), &\ x \ge b \end{cases} $$ Consider the function sequence $f_n$, $n$ $\in$ $\mathbb{N}$,

$$ f_n(x) =\frac{n}{2} ( \bar{f}(x+ \frac{1}{n})-\bar{f}(x- \frac{1}{n})). $$

Show that: $$ \int_{[a,b]} \liminf_{n\rightarrow \infty} f_n(x) d\lambda(x)\le \lim_{\varepsilon\searrow0}(f(b-\varepsilon)-f(a+\varepsilon)) $$

My attempt:

Consider $ \int_{[a,b]} \liminf_{n\rightarrow \infty} f_n(x) d\lambda(x) $. We use the Fatou-Lebesgue theorem. That's means: $ \int_{[a,b]} \liminf_{n\rightarrow \infty} f_n(x) d\lambda(x) $ $\le$ $ \liminf_{n\rightarrow \infty} \int_{[a,b]} f_n(x) d\lambda(x) $ = $ \liminf_{n\rightarrow \infty} \int_{[a,b]} \frac{n}{2} ( \bar{f}(x+ \frac{1}{n})-\bar{f}(x- \frac{1}{n})) d\lambda(x) $ = $\liminf_{n\rightarrow \infty} \int_{[a,b]} \frac{ \bar{f}(x+ \frac{1}{n})-\bar{f}(x- \frac{1}{n})}{(x+\frac{1}{n})-(x-\frac{1}{n})} d\lambda(x) $ = $ \liminf_{n\rightarrow \infty} (\int_{a}^{a+\frac{1}{n}} \frac{ \bar{f}(x+ \frac{1}{n})-\bar{f}(x- \frac{1}{n})}{(x+\frac{1}{n})-(x-\frac{1}{n})}d\lambda(x) +\int_{a+\frac{1}{n}}^{b-\frac{1}{n}} \frac{ \bar{f}(x+ \frac{1}{n})-\bar{f}(x- \frac{1}{n})}{(x+\frac{1}{n})-(x-\frac{1}{n})}d\lambda(x) + \int_{b-\frac{1}{n}}^{b} \frac{ \bar{f}(x+ \frac{1}{n})-\bar{f}(x- \frac{1}{n})}{(x+\frac{1}{n})-(x-\frac{1}{n})}d\lambda(x) ) $

We also know that $ \frac{ \bar{f}(x+ \frac{1}{n})-\bar{f}(x- \frac{1}{n})}{(x+\frac{1}{n})-(x-\frac{1}{n})} $ convergences because $f$ is $\lambda$-almost everywhere differentiable.

Is this correct? I think so. And if is: Please help me to finish this exercise.

Answer 1

Write $$ \int_{[a,b]} \frac{n}{2} ( \bar{f}(x+ \frac{1}{n})-\bar{f}(x- \frac{1}{n})) d\lambda(x)= \frac{n}{2}\int_{[a,b]} \bar{f}(x+ \frac{1}{n})d\lambda(x)-\int_{[a,b]} \bar{f}(x- \frac{1}{n})) d\lambda(x)$$ and then use the change of variables $y=x+\frac1n$ in the first integral and $y=x-\frac1n$ in the second. Then cancel the common parts in those two integrals.