Math Problems Vault

Problem 8.7.1

(a)(b)

Prove the following simple claims.

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(a)

Let $\left\{ f_{n}\right\}_{n=1}^{\infty }$ denote a Cauchy sequence in $L^{p}$ , where $0<p<\infty$ . Show that $\left\{ f_{n}\right\}_{n=1}^{\infty }$ is a Cauchy sequence in measure: for every $\varepsilon >0$ and $\delta >0$ there is $N \in \mathbb {N}$ such that for every $n, m \geqslant N$ we have $\mu \left(\left\{ \omega \in \Omega :\left|f_{n}(\omega )-f_{m}(\omega )\right|>\delta \right\} \right)<\varepsilon$ .

(b)

Let $f, f_{1}, f_{2}, \ldots \in L^{p}$ and suppose that $f_{n} \xrightarrow {L^{p}} f$ , where $0<p<\infty$ . If $g \in L^{\infty }$ , then $f g, f_{1} g, f_{2} g, \ldots \in L^{p}$ and $f_{n} g \xrightarrow {L^{p}} f g$ .

Problem 8.7.2

While convergence in pth mean implies convergence in measure, it is not the case that convergence in measure implies convergence in pthe mean.

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Problem 8.7.3

It is possible for a sequence $\left\{ f_{n}\right\}_{n=1}^{\infty }$ in $L^{p}$ to converge $\mu$ -a. e. to some $f \in L^{p}$ but not in pth mean. That is, convergence $\mu$ -a. e. does not force convergence in $p$ th mean.

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Problem 8.7.4

It is possible for a sequence $\left\{ f_{n}\right\}_{n=1}^{\infty }$ in $L^{p}$ to converge in pth mean to zero, but $\left\{ f_{n}\right\}_{n=1}^{\infty }$ converges at no point of $\Omega$ .

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Problem 8.7.5

It is possible to have functions $f, f_{1}, f_{2}, \ldots \in L^{p_{1}} \cap L^{p_{2}}$ such that $f_{n} \xrightarrow {L^{p_{1}}} f$ but $f_{n} \not\xrightarrow {L^{p_{2}}} f$ .

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Problem 8.7.6

Let $1<p_{0}<+\infty$ . There is a sequence $\left\{ f_{n}\right\}_{n=1}^{\infty }$ in $L^{p}$ for all $1 \leqslant p<p_{0}$ with $f_{n} \xrightarrow {L^{p}} 0$ for all $1 \leqslant p<p_{0}$ , but $f_{n}$ fails to converge in $p_{0}$ th mean.

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Problem 8.7.7

Let $1 \leqslant p, q<+\infty$ , and let $\left\{ f_{n}\right\}_{n=1}^{\infty }$ denote a sequence in $L^{p} \cap L^{q}$ . Suppose that there is $g \in L^{p}$ and $h \in L^{q}$ with $f_{n} \xrightarrow {L^{p}} g$ and $f_{n} \xrightarrow {L^{q}} h$ . Then $g=h$ $\mu$ -a.e. on $\Omega$ .

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Problem 8.7.8

Suppose that $\left\{ f_{n}\right\}_{n=1}^{\infty }$ is a sequence in $L^{p}$ , where $1<p<+\infty$ . Suppose that $h \in L^{1}$ with $f_{n} \leqslant h$ $\mu$ -a.e. on $\Omega$ . If $f_{n}$ converges in measure to some $f$ , then $f \in L^{p}$ and $f_{n} \xrightarrow {L^{p}} f$ .

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Problem 8.7.9

If $f, f_{1}, f_{2}, \ldots \in L^{p}$ are such that $f_{n} \rightarrow f$ uniformly, then $f_{n} \xrightarrow {L^{p}} f$ .

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Remark.

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Here $0<p<+\infty$ .

Problem 8.7.10

Let $f, f_{1}, f_{2}, \ldots \in L^{p}$ , where $1 \leqslant p \leqslant +\infty$ , and suppose that $\left\| f_{n}\right\|_{p} \rightarrow \left\| f\right\|_{p}$ . Then either of the conditions $f_{n} \rightarrow f$ $\mu$ -a.e. on $\Omega$ and $f_{n} \rightarrow f$ in measure is sufficient for the conclusion $f_{n} \xrightarrow {L^{p}} f$ .

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Problem 8.7.11

Consider the measure space $\left(\left[0,1\right], \mathscr {B}, \lambda \right)$ , where here $\mathscr {B}$ denotes the Borel subsets of $\left[0,1\right]$ and $\lambda$ denotes Lebesgue measure restricted to $\mathscr {B}$ . Let $f_{n}=n \; \mathbb {1}_{\left[0, n^{-1}\right]}$ for each $n \in \mathbb {N}$ . Then each $f_{n}$ is in $L^{p}$ for $1 \leqslant p<+\infty$ , $f_{n} \rightarrow 0$ $\mu$ -a.e. on $\left[0,1\right]$ and in measure, but $\int f_{n} g \not\rightarrow \int 0 g$ when $g=1$ , and hence $f_{n} \not\rightarrow 0$ weakly.

Definition. Consider a generic measure space $\left(\Omega , \mathcal{F}, \mu \right)$ , and let $1 \leqslant p<+\infty$ . Let $f, f_{1}, f_{2}, \ldots \in L^{p}$ . We will say that $f_{n} \rightarrow f$ weakly [in $L^{p}$ ] iff

\lim _{n} \int f_{n} g = \int f g \quad \text{for every } g \in L^{p^{\prime }}.

Here $p^{\prime }$ denotes the conjugate exponent of $p$ : $p^{\prime }=\frac{p}{p-1}$ for $1<p<+\infty$ , and $1^{\prime }=+\infty$ .

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Problem 8.7.12

Consider the measure space $\left(\mathbb {R}, \mathscr {B}, \lambda \right)$ . Let $f_{n}=n^{-1} \; \mathbb {1}_{\left[1, e^{n}\right]}$ for every $n \in \mathbb {N}$ . Then each $f_{n}$ is in $L^{p}$ for $1 \leqslant p<+\infty$ , $f_{n} \rightarrow 0$ uniformly, but $f_{n} \not\rightarrow 0$ weakly in $L^{p}$ .

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Remark.

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Consider the function $g(\omega )=\omega^{-1} \; \mathbb {1}_{[1,+\infty )}(\omega )$ . We have that $g \in L^{p^{\prime }}$ for each $1<p<+\infty$ , $g$ is bounded, and $\int f_{n} g=1$ for every $n \in \mathbb {N}$ , while $\int 0 g=0$ .

Problem 8.7.13

Consider the measure space $\left(\left[0,2 \pi \right], \mathscr {B}, \lambda \right)$ , where here $\mathscr {B}$ denotes the Borel subsets of $\left[0,2 \pi \right]$ and $\lambda$ denotes Lebesgue measure restricted to $\mathscr {B}$ . For each $n \in \mathbb {N}$ and $0 \leqslant \omega \leqslant 2 \pi$ , define $f_{n}(\omega )=\cos n \omega$ . Then each $f_{n}$ is in $L^{1}$ , and $f_{n} \rightarrow 0$ weakly in $L^{1}$ . However, $f_{n} \not\rightarrow 0$ . In fact, we have $f_{n} \not\xrightarrow {\mu } 0$ and $f_{n} \not\rightarrow 0$ $\mu$ -a.e. on $\Omega$ .

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Remark.

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This exercise might be difficult.

Problem 8.7.14

Refer to the Vitali Convergence Theorem. Prove that conditions (i) and (ii) imply that $f \in L^{p}$ and $f_{n} \rightarrow f$ in pth mean when $\mu (\Omega )=+\infty$ .

Vitali Convergence Theorem (finite-measure case). Let $\left(\Omega , \mathcal{F}, \mu \right)$ denote a measure space with $\mu (\Omega )<+\infty$ , and let $1 \leqslant p<+\infty$ . Let $\left\{ f_{n}\right\}_{n=1}^{\infty }$ denote a sequence of functions in $L^{p}$ . Let $f$ denote an $\mathcal{F} / \mathscr {B}^{*}$ -measurable function on $\Omega$ such that $f_{n} \rightarrow f$ $\mu$ -a.e. on $\Omega$ and $f$ is finite $\mu$ -a.e. on $\Omega$ . Then the condition $f \in L^{p}$ and $f_{n} \xrightarrow {L^{p}} f$ is equivalent to the dual conditions (i) for every $\varepsilon >0$ there is $A \in \mathcal{F}$ [depending on $\varepsilon$ ] with $\mu (A)<+\infty$ and $\int_{A^{c}} \left|f_{n}\right|^{p}<\varepsilon$ for every $n \in \mathbb {N}$ , and (ii) for every $\varepsilon >0$ there is $\delta >0$ [depending only on $\varepsilon$ , not $n$ ] such that whenever $\mu (E)<\delta$ we have $\int_{E} \left|f_{n}\right|^{p}<\varepsilon$ for every $n \in \mathbb {N}$ .

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Problem 8.7.15

Let $\left(\Omega , \mathcal{F}, \mu \right)$ denote a measure space with $1 \leqslant p<+\infty$ . Let $\left\{ f_{n}\right\}_{n=1}^{\infty }$ denote a sequence in $L^{p}$ , and let $f$ denote an $\mathcal{F} / \mathscr {B}^{*}$ -measurable function that is finite $\mu$ -a.e. on $\Omega$ with $f_{n} \rightarrow f$ $\mu$ -a.e. on $\Omega$ . For every $n, k \in \mathbb {N}$ , let $B_{n, k}=\left\{ \omega \in \Omega :\left|f_{n}(\omega )\right|^{p} \geqslant k\right\}$ . Make the following additional assumption (a): for every $\varepsilon >0$ there is a set $A \in \mathcal{F}$ [depending on $\varepsilon$ ] such that $\mu (A)<+\infty$ and $\int_{A^{c}} \left|f_{n}\right|^{p}<\varepsilon$ for every $n \in \mathbb {N}$ .

Then the following four statements are logically equivalent:

$f \in L^{p}$ and $f_{n} \xrightarrow {L^{p}} f$
If $\left\{ E_{k}\right\}_{k=1}^{\infty }$ is a nonincreasing sequence of $\mathcal{F}$ -sets with limit $\varnothing$ , then for every $\varepsilon >0$ there is $K \in \mathbb {N}$ such that $k \geqslant K$ forces $\int_{E_{k}} \left|f_{n}\right|^{p}<\varepsilon$ for every $n \in \mathbb {N}$ .
For every $\varepsilon >0$ there is $K \in \mathbb {N}$ such that $k \geqslant K$ forces $\int_{B_{n, k}} \left|f_{n}\right|^{p}<\varepsilon$ for every $n \in \mathbb {N}$ .
For every $\varepsilon >0$ there is $\delta >0$ such that $\mu (E)<\delta$ forces $\int_{E} \left|f_{n}\right|^{p}<\varepsilon$ for every $n \in \mathbb {N}$ .

Vitali Convergence Theorem (finite-measure case). Let $\left(\Omega , \mathcal{F}, \mu \right)$ denote a measure space with $\mu (\Omega )<+\infty$ , and let $1 \leqslant p<+\infty$ . Let $\left\{ f_{n}\right\}_{n=1}^{\infty }$ denote a sequence of functions in $L^{p}$ . Let $f$ denote an $\mathcal{F} / \mathscr {B}^{*}$ -measurable function on $\Omega$ such that $f_{n} \rightarrow f$ $\mu$ -a.e. on $\Omega$ and $f$ is finite $\mu$ -a.e. on $\Omega$ . Then the condition $f \in L^{p}$ and $f_{n} \xrightarrow {L^{p}} f$ is equivalent to the dual conditions (i) for every $\varepsilon >0$ there is $A \in \mathcal{F}$ with $\mu (A)<+\infty$ and $\int_{A^{c}} \left|f_{n}\right|^{p}<\varepsilon$ for every $n \in \mathbb {N}$ , and (ii) for every $\varepsilon >0$ there is $\delta >0$ such that whenever $\mu (E)<\delta$ we have $\int_{E} \left|f_{n}\right|^{p}<\varepsilon$ for every $n \in \mathbb {N}$ .

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Remark.

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The following outline is useful. By the Vitali Convergence Theorem and Exercise 8.7.14, it is immediate that (i) and (iv) are logically equivalent.

We show that (i) implies (ii). Let $\varepsilon >0$ . Find $N \in \mathbb {N}$ such that $n \geqslant N$ implies $\left\| f_{n}-f\right\|_{p}<\varepsilon$ . For $n \geqslant N$ and $k \in \mathbb {N}$ , Minkowski's Inequality gives

\left(\int _{E_{k}} \left|f_{n}\right|^{p}\right)^{1 / p} \leqslant \left(\int _{E_{k}} \left|f\right|^{p}\right)^{1 / p}+\left(\int _{E_{k}} \left|f_{n}-f\right|^{p}\right)^{1 / p} <\left(\int _{E_{k}} \left|f\right|^{p}\right)^{1 / p}+\varepsilon .

Apply DCT to $\left\{ \left|f\right|^{p} \; \mathbb {1}_{E_{k}}\right\}_{k=1}^{\infty }$ to show that $\left(\int_{E_{k}} \left|f_{n}\right|^{p}\right)^{1 / p}<\varepsilon$ for large enough $k$ , say $k \geqslant K$ . Then for every $n \in \mathbb {N}$ and $k \geqslant K$ , we have $\left(\int_{E_{k}} \left|f_{n}\right|^{p}\right)^{1 / p}<2 \varepsilon$ . For $n \in \left\{ 1, \ldots , N-1\right\}$ , observe for every $k \in \mathbb {N}$ that we have

\int _{E_{k}} \left|f_{n}\right|^{p} \leqslant \int _{E_{k}} \max \left\{ \left|f_{1}\right|^{p}, \ldots , \left|f_{N-1}\right|^{p}\right\} <+\infty ,

and therefore DCT can be used to obtain (ii).

Now assume (ii), picking $\varepsilon >0$ . For each $k \in \mathbb {N}$ , define $E_{k}=\bigcup_{n=k}^{\infty } B_{n, k}$ . Then $E_{1} \supseteq E_{2} \supseteq \cdots$ and $\overline{\lim }_{n} \left|f_{n}(\omega )\right|=+\infty$ on $\bigcap_{k=1}^{\infty } E_{k}$ ; from this we have $\mu \left(\bigcap_{k=1}^{\infty } E_{k}\right)=0$ . Next, write $F_{k}=E_{k}-\bigcap_{j=1}^{\infty } E_{j}$ for every $k \in \mathbb {N}$ . We have that $F_{1} \supseteq F_{2} \supseteq \cdots$ and $\lim_{k} F_{k}=\varnothing$ . Choose $K$ such that $k \geqslant K$ implies $\int_{F_{k}} \left|f_{n}\right|^{p}<\varepsilon$ for every $n \in \mathbb {N}$ . For every $n \geqslant K$ we have $B_{n, K} \subseteq E_{K}$ . Therefore, if $n, k \geqslant K$ , then

\int _{B_{n, k}} \left|f_{n}\right|^{p} \leqslant \int _{B_{n, K}} \left|f_{n}\right|^{p} \leqslant \int _{E_{K}} \left|f_{n}\right|^{p}=\int _{F_{K}} \left|f_{n}\right|^{p}<\varepsilon .

For $n \in \left\{ 1, \ldots , K-1\right\}$ , we have $\left|f_{n}\right|^{p} \leqslant \max \left\{ \left|f_{1}\right|^{p}, \ldots , \left|f_{K-1}\right|^{p}\right\} =g$ , say, and hence $\int_{B_{n, k}} \left|f_{n}\right|^{p} \leqslant \int_{B_{k}} g<+\infty$ , where $B_{k}=\bigcup_{n=1}^{K-1} B_{n, k}=\left\{ \omega \in \Omega : g(\omega ) \geqslant k\right\}$ . Use DCT to derive (iii) in direct fashion.

Now assume (iii). Pick $K \in \mathbb {N}$ such that $k \geqslant K$ forces $\int_{B_{n, k}} \left|f_{n}\right|^{p}<\varepsilon^{p}$ for every $n \in \mathbb {N}$ . Argue that $\mu (E)<K^{-1} \varepsilon^{p}$ implies

\left(\int _{E} \left|f_{n}\right|^{p}\right)^{1 / p} \leqslant \left(\int _{E \cap B_{n, K}} \left|f_{n}\right|^{p}\right)^{1 / p}+\left(\int _{E \cap B_{n, K}^{c}} \left|f_{n}\right|^{p}\right)^{1 / p}<2 \varepsilon .

Therefore we may deduce (iv) from (iii).

Problem 8.7.16

Assume everything in Exercise 8.7.15, including (a). Assume further that (ii) holds. Argue that (i) and (ii) of the Vitali Convergence Theorem follow as consequences.

Vitali Convergence Theorem (finite-measure case). Let $\left(\Omega , \mathcal{F}, \mu \right)$ denote a measure space with $\mu (\Omega )<+\infty$ , and let $1 \leqslant p<+\infty$ . Let $\left\{ f_{n}\right\}_{n=1}^{\infty }$ denote a sequence of functions in $L^{p}$ . Let $f$ denote an $\mathcal{F} / \mathscr {B}^{*}$ -measurable function on $\Omega$ such that $f_{n} \rightarrow f$ $\mu$ -a.e. on $\Omega$ and $f$ is finite $\mu$ -a.e. on $\Omega$ . Then the condition $f \in L^{p}$ and $f_{n} \xrightarrow {L^{p}} f$ is equivalent to the dual conditions (i) for every $\varepsilon >0$ there is $A \in \mathcal{F}$ with $\mu (A)<+\infty$ and $\int_{A^{c}} \left|f_{n}\right|^{p}<\varepsilon$ for every $n \in \mathbb {N}$ , and (ii) for every $\varepsilon >0$ there is $\delta >0$ such that whenever $\mu (E)<\delta$ we have $\int_{E} \left|f_{n}\right|^{p}<\varepsilon$ for every $n \in \mathbb {N}$ .

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Remark.

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The particular form of the Vitali Convergence Theorem as presented in this text and the previous two exercises [and the outlines of the solution] owe their existence and form to the most excellent presentation of Hewitt and Stromberg, pp. 203-205.