Counterexamples and convergences

Question

I want to find counterexamples for the following statements:

• If $\int f_n\to \int f$ then $\int |f_n-f|\to 0$.
• If $\int |f_n-f|\to 0$ then $f_n\to f$ almost everywhere.

Can you give me a hint of what examples may work?

Answer 1

For the first part:

Define $f_n = 1_{[0,1]}$ for all $n$ (this is a constant sequence), and let $f=1_{[1,2]}$. Then $\int f_n =1 = \int f$ for all $n$, but $\int |f_n - f| =2$ for all $n$.

For the second part:

Take some enumeration $\{I_n\}$ of all subintervals of $[0,1]$ which have the form $[\frac{k}{m},\frac{k+1}{m}]$ for some integers $m,k$.

Let $f_n=1_{I_n}$, so $\int |f_n-0| = m(I_n) \to 0$, because for any $\epsilon >0$, there are only finitely many $I_n$ with $m(I_n)>\epsilon$. However, for any $x \in [0,1]$ there are infinitely many $n$ with $f_n(x)=1$, so $f_n$ does not converge a.e. to $0$.

Answer 2

The following two counterexamples take place in $ [0,1] $ with its standard $ \sigma $-algebra and Lebesgue measure.

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Counterexample for the first:

Let $ (f_{n})_{n \in \Bbb{N}} \stackrel{\text{df}}{=} \left( \chi_{[n,n + 1]} \right)_{n \in \Bbb{N}} $ and $ f \stackrel{\text{df}}{=} \chi_{[0,1]} $.

Counterexample for the second:

Let $ H_{n} $ denote the $ n $-th harmonic number, and for each $ n \in \Bbb{N} $, let $ A_{n} $ denote the translation of the points in the interval $ [H_{n},H_{n + 1}] $ back to the interval $ [0,1] $. Then let $ (f_{n})_{n \in \Bbb{N}} \stackrel{\text{df}}{=} (\chi_{A_{n}})_{n \in \Bbb{N}} $ and $ f \stackrel{\text{df}}{\equiv} 0 $.

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Additional information

Concerning the second counterexample, observe that by the divergence of the harmonic series, each point in $ [0,1] $ appears in $ A_{n} $ for infinitely many $ n \in \mathbb{N} $. Hence, $ (\chi_{A_{n}})_{n \in \Bbb{N}} $ does not converge to the zero-function anywhere in $ [0,1] $. However, $ \mu(A_{n}) = \dfrac{1}{n} $ for each $ n \in \mathbb{N} $, so $$ \lim_{n \to \infty} \int_{[0,1]} \chi_{A_{n}} ~ \mathrm{d}{\mu} = \lim_{n \to \infty} \mu(A_{n}) = 0. $$

Answer 3

Some examples on $[0,1]$: For the first one, let $f_n(x)=\cos nx, f(x) = 0.$ For the second one, consider the characteristic functions, in order, of $[0,1],[0,1/2],[1/2,1],[0,1/3],[1/3,2/3],[2/3,1], \dots.$