Necessity of hypotheses (a theorem fails without them)

Give an example to show that the Monotone Convergence Theorem (3.11) can fail if the hypothesis that $f_{1}, f_{2}, \ldots$ are nonnegative functions is dropped.

Give an example to show that the Monotone Convergence Theorem can fail if the hypothesis of an increasing sequence of functions is replaced by a hypothesis of a decreasing sequence of functions.

Let $f : \mathbb {R}^{n} \setminus \left\{ 0\right\} \rightarrow \mathbb {R}$ be a function which is continuously differentiable and whose partial derivatives are uniformly bounded:

for all $(x_1, \ldots , x_n) \neq (0, \ldots , 0)$. Show that if $n \geq 2$, then $f$ can be extended to a continuous function defined on all of $\mathbb {R}^{n}$. Show that this is false if $n = 1$ by giving a counterexample.

Let $p>1$ and $a_{n} \downarrow 0$. Prove that the convergence of the series $\sum a_{n}^{p-1} n^{-1 / p}$ implies the convergence of the series $\sum a_{n}^{p}$. Is monotonicity essential?

Prove that Proposition 6.3.8 is not true if $\mu (X)=\infty$.

Proposition 6.3.8: Let $(X, \mathfrak {M}, \mu )$ be a finite measure space and let $\left\{ f_{n}\right\}_{n=1}^{\infty }$, $f$ be measurable functions on $X$ with values in $[-\infty , +\infty ]$ that are finite a.e. If $\left\{ f_{n}\right\}$ converges to a real-valued function $f$ a.e. on $X$, then $\left\{ f_{n}\right\}$ converges to $f$ in measure on $X$.

Prove that Egorov's Theorem need not hold if $\mu (X)=\infty$.

Show that the Monotone Convergence Theorem may not hold for decreasing sequences of functions.

Give an example that the Monotone Convergence Theorem does not hold without the assumption that each $f_{n}$ is nonnegative.

The purpose of this exercise is to show that in the absence of $\sigma$ finiteness the product measure need not be unique. Let $\Delta =\left\{ (x, y) \in [0,1]^{2}: x=y\right\}$, let $m$ be the Lebesgue measure on $\mathcal{B}([0,1])$, let $\nu$ be the counting measure on $\mathbb {P}([0,1])$, and let $m \times \nu$ be the Carathéodory extension of the measure $\omega$ defined on measurable rectangles by $\omega (E \times F)=m(E) \nu (F)$. Define $\lambda$ on $\mathcal{B}([0,1]) \otimes \mathbb {P}([0,1])$ by $\lambda (E)=\sup \left\{ \sum_{y \in A} m_{y}(([0,1] \times \left\{ y\right\} ) \cap E): A \text{ is a finite subset of } [0,1]\right\}$, where $m_{y}$ is the Lebesgue measure on $[0,1] \times \left\{ y\right\}$. Prove that:

Let $f$ and $g$ be nonnegative and continuous at $c$.

Give an example to show that the conclusions of Exercise 9.1.21 need not hold if the continuity assumption on $f$ is dropped.

Show that the Vitali Covering Lemma does not extend to the case in which the covering collection contains some intervals that are not necessarily nondegenerate.

Show that the condition imposed on the set $A$ in Exercise 9.4.37 (a) is not necessary, i.e., that there is a set that does not satisfy it, yet $F(x):= m([0, x] \cap A)$ is an absolutely continuous, strictly increasing function on $[0,1]$ such that $F^{\prime }=0$ on a set of positive measures.

Give an example to show that Proposition 9.5.3 fails if $\nu$ is a $\sigma$-finite measure.

Proposition 9.5.3: Let $(X, \mathfrak {M}, \mu )$ be a measure space and $\nu$ a finite measure on $(X, \mathfrak {M})$. Then $\nu$ is absolutely continuous with respect to $\mu$ if and only if for every $\varepsilon > 0$ there is $\delta > 0$ such that for every $E \in \mathfrak {M}$, $\mu \left(E\right) < \delta$ implies $\nu \left(E\right) < \varepsilon$.

Let $X$ be an uncountable set, and let $\mathfrak {M}$ consist of those sets $A$ such that either $A$ or $A^{C}$ is countable. Define two measures $\mu , \nu$ on $\mathfrak {M}$ by $\mu (A)= \nu (A)=0$ if $A$ is countable, and $\mu (A)=1, \nu (A)=\infty$ if $A^{C}$ is countable.

Give an example that the Radon-Nikodym theorem fails when $\nu$ is not $\sigma$-finite.

Prove or disprove: The Lebesgue Decomposition Theorem remains true if measures $\mu$ and $\nu$ are not assumed to be $\sigma$-finite.

Show that Theorem 2.36 and its Corollary become false (in $\mathbb {R}^{1}$, for example) if the word "compact" is replaced by "closed" or by "bounded".

Find continuous functions $f_{n}:[0,1] \rightarrow [0, \infty )$ such that $f_{n}(x) \rightarrow 0$ for all $x \in [0,1]$ as $n \rightarrow \infty$, $\int_{0}^{1} f_{n}(x) d x \rightarrow 0$, but $\sup_{n} f_{n}$ is not in $L^{1}$. (This shows that the conclusion of the dominated convergence theorem may hold even when part of its hypothesis is violated).

Let $\Omega =\left\{ \omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}\right\}$, let $\mathcal{A}$ consist of $\varnothing ,\left\{ \omega_{1}, \omega_{2}\right\}$, $\left\{ \omega_{1}, \omega_{3}\right\} ,\left\{ \omega_{2}, \omega_{4}\right\} ,\left\{ \omega_{3}, \omega_{4}\right\}$, and $\Omega$, and let $\mu : \mathcal{A} \rightarrow \mathbb {R}$ be defined as follows: $\mu (\Omega )=6, \mu (\varnothing )=0$, and $\mu \left(\left\{ \omega_{1}, \omega_{2}\right\} \right)=\mu \left(\left\{ \omega_{1}, \omega_{3}\right\} \right)=\mu \left(\left\{ \omega_{2}, \omega_{4}\right\} \right)=\mu \left(\left\{ \omega_{3}, \omega_{4}\right\} \right)=$ 3.

Show that the result in Exercise 12 can fail without the hypothesis that $V$ is finite-dimensional.

Prove that the condition of monotonic nondecrease of the function $f$ in Problem 524 is essential. If it is dropped, the statement may fail. $\circledast$

Find counterexamples to show that Theorem 1.25 is incorrect if we remove the requirements that $u$ must be a nonfinal vertex and $v$ must be a noninitial vertex.

Theorem 1.25: Let $r$ be a regular expression. Then, an $\varepsilon$-edge $(u, v)$ in $G(r)$ which is a unique out-edge from a nonfinal vertex $u$ or a unique in-edge to a noninitial vertex $v$ can be shrunk into a single vertex, still preserving the property of Theorem 1.23. (If one of the endpoints of the $\varepsilon$-edge is the initial vertex or the final vertex, then so is the resulting vertex.)

Prove that the new initial state $s$ and the final state $f$ in the construction of Example 2.22 are necessary. That is, find NFA's $M_{1}$ and $M_{2}$ such that the NFA's $M_{1}^{\prime }$ and $M_{2}^{\prime }$ of Figure 2.28 have the property $L\left(M_{1}^{\prime }\right) \neq L\left(M_{1}\right)^{*}$ and $L\left(M_{2}^{\prime }\right) \neq L\left(M_{2}\right)^{*}$.

Give a counterexample to show that the following construction fails to prove Theorem 1.49, the closure of the class of regular languages under the star operation. ¹ Let $N_{1}=\left(Q_{1}, \Sigma , \delta_{1}, q_{1}, F_{1}\right)$ recognize $A_{1}$. Construct $N=\left(Q_{1}, \Sigma , \delta , q_{1}, F\right)$ as follows. $N$ is supposed to recognize $A_{1}^{*}$.

Give a counterexample to show that the following construction fails to prove that the class of context-free languages is closed under star. Let $A$ be a CFL that is generated by the CFG $G=(V, \Sigma , R, S)$. Add the new rule $S \rightarrow S S$ and call the resulting grammar $G^{\prime }$. This grammar is supposed to generate $A^{*}$.