Monday September 21:
- Work:
- Show that $E\subset X$ is totally bounded if and only if
there is an $\epsilon$-net for $E$ for all $\epsilon >0$.
- Suppose that $(X,\rho)$ is compact and that $f:(X,\rho)\to
(Y,\sigma)$ is continuous. Show that $f(X)$ is compact in
$Y$.
- Let $X=(0,1)$. For each $x\in X$, let
$B_{\delta_x}(x)=\{\,y\in (0,1):|x-y|< \delta\,\}$ be such that
$y\in B_{\delta_{x}}(x)$ implies
$\bigl|\frac1x-\frac1y\bigr|<1$. Show that the cover
$$(0,1)=\bigcup_{x\in(0,1)} B_{\delta_x}(x)$$ has no Lebesgue
number.
- Show that a compact metric space has a countable dense
subset. (Actually, it is enough for the space to be totally
bounded.)
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Wednesday:
- Work:
- Let $\mathcal F$ be the family of functions $f_n(x)=x^n$
on $X=[0,1]$. Show that $\mathcal F$ is equicontinuous at
each $x\in [0,1)$. (Luke, invoke the force in the form of the
Mean Value Theorem.)
- Show that an equicontinuous family of functions on a
compact metric space is uniformly equicontinuous as stated in
lecture. (Some texts do not define equicontinuous at a point.
Instead, whether $X$ is compact or not, equicontinuity is
what we have called uniformly equicontinuity. Fortunately,
there is no distinction for compact spaces.)
- Show that a compact subspace of a metric space must be
closed. Conversely, show that a closed subspace of a compact
metric space is compact.
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Friday:
- Homework Solutions: Here
are selected solutions for
the homework. (Last modified December 31, 1969)
- Work:
- Let $K$ be a compact subset of a metric space $X$ and let
$K \subset U$ with $U$ open. Show that there is an open set $V$
such that $K\subset V\subset \overline V\subset U$.
(Consider homework problem #10.)
- (In this problem, we will assume that if $(X,\rho)$ and
$(Y,\sigma)$ are metric spaces then so is $(X\times
Y,\delta)$ where
$\delta((x,y),(x',y'))=\rho(x,x')+\sigma(y,y')$. You can
also assume that with respect to this product metric,
$(x_n,y_n)\to (x,y)$ if and only if $x_n\to x$ and $y_n\to
y$. In particular, if $(X,\rho)$ and $(Y,\sigma)$ are
complete, so is $(X\times Y,\delta)$.) Let $U$ be a nonempty
open subset of a complete metric space $(X,\rho)$. Show that
$U$ admits a complete metric which is equivalent to that
inherited from $X$. I suggest the following.
- It suffices to find a homeomorphism
$\phi:(U,\rho)\to (Y,\sigma)$ where $(Y,\sigma)$ is
complete. (Recall that $\phi$ is a
homeomorphism if $\phi$ is continuous, one-to-one and
onto, and such that $\phi^{-1}$ is also
continuous.)
- Let $A=X\setminus U$ and define $f:U\to \mathbf R$
by $f(x)=\rho(x,A)^{-1}$. Then the map $\phi(x)=(x,f(x))$ is
continuous from $(U,\rho)$ to $(X\times\mathbf
R,\delta)$ where $\delta$ is the obvious complete product
metric. It suffices to see that the range of
$\phi$ is closed.
- If $E$ is a subset of a metric space $(X,\rho)$, then its
boundary, $\partial E$ is closed. Show that if $x\in
E\setminus \partial E$, then $x\in \operatorname{Int}(E)$. If $E$ is
closed, show that $\partial E$ has no interior.
- Suppose that $X$ is a complete metric space without
isolated points, and that
$O_n$ is open and dense for all $n\ge1$. Show that
$\bigcap_{n=1}^\infty O_n$ is uncountable. (Note that if $x$
is not isolated, then $X\setminus \{x\}$ is open and
dense.)
- Suppose that $(X,\rho)$ is a compact metric space and that
$f:X\to X$ satisfies $\rho(f(x),f(y))<\rho(x,y)$ whenever
$x\not=y$. Show that
$f$ has a unique fixed point.
- Let $f(x)=\pi/2+x-\arctan(x)$. Show that
$|f(x)-f(y)|<|x-y|$ for all $x,y\in\mathbf{R}$ with $x\not=y$,
but that $f$ does not
have a fixed point. Compare this result to the previous problem.
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Monday, October 5:
- Homework: Turn in problems 11-23 on Wednesday, October
7th via Gradescope. It is unlikely that gradescope will
know about homework #2 prior to Wednesday morning.
- Work:
- Show that the countable union of sets of measure
zero in $\mathbf{R}$ has measure zero. (I am using the term "measure zero"
instead of "content zero" as that will be more relvant down the
road.)
- Suppose $f:[a,b]\to\mathbf{R}$ is bounded, and let $\mathcal P$ and
$\mathcal Q$ be
partitions of $[a,b]$. Prove that $L(f,\mathcal P)\le U(f,\mathcal Q)$, where
$L(f,\mathcal P)$ and
$U(f,\mathcal Q)$ are the lower and upper Riemann sums, respectively, for
$f$ on $[a,b]$. (Hint: the result is trivial if $\mathcal
P=\mathcal Q$; now
let $\mathcal R=\mathcal P\cup\mathcal Q$.)
- Prove that a bounded function $f:[a,b]\to\mathbf{R}$ is Riemann integrable
on $[a,b]$
if and only if for all $\epsilon>0$ there is a partition $\mathcal P$ of
$[a,b]$ such that $$
U(f,\mathcal P)-L(f,\mathcal P)<\epsilon.
$$
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Wednesday, October 7:
- Work:
- Suppose that $(X,\mathcal M)$ is a measurable space. Show
that if $\mathcal M$ is
countable, then $\mathcal M$ is finite. (Hint: since $\mathcal M$
is countable,
you can show that $\omega_{x}=\bigcap\{\,E:\text{$E\in\mathcal M$ and
$x\in E$}\,\}$ belongs to $\mathcal M$. The sets $\{\omega_{x}\}_{x\in X}$
partition $X$.) This is from Rudin's Real & Complex, page
31, #1
- Let $X$ be an uncountable set and let $\mathcal M$ be the
collection of
subsets $E$ of $X$ such that either $E$ or $E^{C}$ is countable.
Prove that $\mathcal M$ is a $\sigma$-algebra.
- Let $(a_{n})$ be a sequence in $[-\infty,\infty]$.
- Show that $\liminf_{n}a_{n}\le \limsup_{n}a_{n}$.
- Suppose that $\lim a_{n}$ exists and equals
$L\in [-\infty,\infty]$. Show that
$\limsup_{n}a_{n}=L=\liminf_{n}a_{n}$.
- Suppose that $\limsup_{n}a_{n}=L=\liminf_{n}a_{n}$. Show that
$\lim_{n}a_{n}$ exists and equals $L$.
- Borel Sets (OPIONAL -- Do not turn in):
If you are
willing to learn about ordinals and transfinite induction (Secton 0 of
Folland's Real Analysis is a good source for this), then you
can see that the cardinality of the Borel sets in $\mathbf R$ is the
same as the same of that of $\mathbf R$ itself (usually written
$\mathfrak c$). In particular, $\mathcal B(\mathbf R)$ is strictly
smaller than $\mathcal P(\mathbf R)$. Moreover, you get some insight
as to what is in the $\sigma$-algebra generated by a subset of
$\mathcal P(\mathbf R)$. I wrote
some directed problems about this some
years ago. If you prefer, here are the
solutions.
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Friday, October 9:
- Homework Solutions: Here
are selected solutions for
the homework. (Last modified December 31, 1969)
- Work:
- Recall from calculus that if $\{a_{n}\}$ is a sequence of
nonnegative real numbers, then $\sum_{n=1}^{\infty} a_{n} = \sup_{n}
s_{n}$, where $s_{n} = a_{1}+\dots+ a_{n}$. (Note the value
$\infty$ is allowed.)
- Show that $\sum_{n=1}^{\infty} a_{n} =\sup\{\,\sum_{k\in
F}a_{k}: \text{$F$ is a finite subset of
$\mathbf{N}=1,2,3,\dots$}\,\}$. (The point of this problem
is that if $I$ is a (not necessarily countable) set, and if
$a_{i}\ge0$ for all $i\in I$, then we can define $\sum_{i\in
I} a_{i} = \sup\{\,\sum_{k\in F}a_{k}: \text{$F$ is a finite
subset of $I$}\,\}$, and our new definition coincides with the
usual one when both make sense.)
- Now let $X$ be a set and $f:X\to[0,\infty)$ a function. For
each $E\subset X$, define
\begin{equation*}
\nu(E) := \sum_{x\in E} f(x).
\end{equation*}
Show that $\nu$ is a measure on $\bigl(X,\mathcal P(X)\bigr)$. (Note
that some care is required here. We can't pass operations like
supremums and limits though
infinite sums without justification! Also note that in
lecture, we
considered the special cases of counting measure, where
$f(x)=1$ for all $x\in X$, and the delta measure at $x_{0}$, where
$f(x_{0})=1$ for some $x_{0}\in X$ and $f(x)=0$ otherwise. Another
important example is the case where $\sum_{x\in X}f(x)=1$. Then $f$
is a (discrete) probability distribution on $X$ and $\nu(E)$ is the
probability of the event $E$ for this distribution.)
- Let $X$, $f$, and $\nu$ be as in part (2). Show that if
$\nu(E)<\infty$, then $\{\,x\in E:f(x)>0\,\}$ is countable. Hint:
if $\{\,x\in E:f(x)>0\,\}$ is uncountable, then for some $m\in
\mathbf{N}$, the set $\{\,x\in E:f(x)>\frac1m\,\}$ is infinite.
(Note that this last result says that discrete probability
distributions ``live on'' countable sample spaces.)
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Monday, October 12:
- Work:
- Suppose that
$f,g:(X,\mathcal M)\to[-\infty,\infty]$ are measurable functions. Prove
that the sets $$
\{\,x:f(x)< g(x)\,\}\quad\text{and}\quad\{\,x:f(x)=g(x)\,\}$$
are measurable. (Remark: if $h=f-g$ were defined, then this problem
would be much easier (why?). The problem is that $\infty-\infty$ and
$-\infty+\infty$ make no sense, so $h$ may not be everywhere defined.)
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Friday, October 16:
- A Cheat: To construct some counterexamples below,
let's assume that we have defined Lebesgue $m$ on $(\mathbf R,\mathcal
B(\mathcal R))$ so that the Lebesgue integral conicides with the
Riemann integral if $f$ is Riemann integrable. Thus you can
assume $1_{[-n,n]}$ has integral $2n$ or that $1_{[0,\infty)}$
has infinite integral.
- Work:
- Let $X$ be an uncountable set and $\mathcal M$ the
$\sigma$-algebra from problem #28 above. Define $\mu:\mathcal
M\to [0,\infty]$ by $\mu(E)=0$ if $E$ is countable and
$\mu(E)=1$ if $E$ is uncountable.
- Show that $\mu$ is a measure on $(X,\mathcal M)$.
- Describe the measurable functions $f:X\to\mathbf C$ and
their integrals. (Hint: show that a measurable function
must be constant off a countable set; that is, $f$ must be
constant $\mu$-almost everywhere.)
- Suppose that $f_n:X\to\mathbf R$ is measurable for all
$n\in \mathbf N$.
- Show that it need not be the case that
$$\int_X \liminf_n f_n \,d\mu\le \liminf_n \int_X
f_n\,d\mu.$$
That is, the conclusion of Fatou's Lemma is false if we
no not assume the $f_n$ are non-negative.
- Suppose now that $f_n\le f_{n+1}$ for each $n\in\mathbf
N$. Show that it need not be the case that
$$\lim_n\int_X f_n\,d\mu=\int_X f\,d\mu.$$ That is, the
conclusion of the Monotone Convergence Theorem can fail
if we don't insist that the $f_n$ are non-negative. What
if we assume $f_1\in\mathcal L^1(X)$?
- Suppose that $(X,\mathcal M,\mu)$ is a measure space with
$\mu(X)<\infty$. (We say that $(X,\mathcal M,\mu)$ is a
finite measure space.) Show that if $f_n:X\to \mathbf C$ is a
sequence of bounded measurable functions converging uniformly
to $f$, then $f\in \mathcal L^1(X)$ and $$\lim_n\int_X
f_n\,d\mu=\int_Xf\,d\mu.$$ Show that the hypothesis that
$\mu(X)<\infty$ can't be ommited.
- Suppse that $f\in \mathcal L^1(X,\mathcal M,\mu)$. Show
that for all $\epsilon >0$ there is a $\delta>0$ such that
$\mu(E)<\delta$ implies that $$\int_E|f|\,d\mu<\epsilon.$$
(Hint: First work the problem assuming that $f$ is bounded.
Then let $E_n =\{\,
x:|f(x)|\le n\,\}$ and $f_n=1_{E_n}\cdot f$. Then observe that
$\int_X|f-f_n|\,d\mu\to 0$.)
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Monday, October 19:
- Work:
- Suppose that $\{A_n\}_{n=1}^\infty$ are subsets of a set
$X$. Let $B_1=A_1$ and $B_n=A_n\setminus
\bigcup_{k=1}^{n-1}A_k$ for $n\ge2$. Show that the $B_n$ are
pairwise disjoint, $B_n \subset A_n$, and that
$\bigcup_{k=1}^nA_k=\bigcup_{k=1}^n
B_k$ for all $n$. Observe that if $(X,\mathcal M)$ is a measurable
space and the $A_n$ are measurable, then so are the $B_n$. (I
sometimes call this process `disjointification'. This really
a joke since I don't believe `disjointification' is really a
word.)
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Wednesday, October 21:
- Work:
- Suppose that $Y$ is a topological space and that $\mathcal
M$ is a $\sigma$-algebra in $Y$ containing all the Borel sets.
Suppose that $\mu$ is a measure on $(Y,\mathcal M)$ such that
for all $E\in\mathcal M$ $$ \mu(E)=\inf\{\,\mu(V):\text{$V$ is
open and $E\subset V$}\,\}. $$ Suppose also that
$$Y=\bigcup_{n=1}^\infty Y_n\quad\text{with $\mu(Y_n)<\infty$
for all $n\ge1$.}\label{sfin}\tag{$*$}$$ In this case we say
that $\mu$ is a $\sigma$-finite outer-regular measure on
$(Y,\mathcal M)$.
- Show that Lebesgue measure $m$ on $(\mathbf R,\mathcal L)$
is an example of a $\sigma$-finite outer
regular measure.
- If $E\in\mathcal M$ and if $\epsilon>0$, then show that
there is an open set $V$ and a closed set $F$ such that
$F\subset E\subset V$ with $\mu(V\setminus F)<\epsilon$. (Hint:
first assume $\mu(E)<\infty$. Then use \eqref{sfin}.)
- Recall that a countable intesection of open sets is called
a $G_\delta$-set, and that a countable union of closed sets is
called a $F_\sigma$-set. Show that if $E\in \mathcal M$, then
there is a $G_\delta$-set $G$ and a $F_\sigma$-set $A$ such
that $A\subset E\subset G$ and $\mu(G\setminus F)=0$.
- Use the above to conclude that $(\mathbf R,\mathcal L,m)$ is the
completion of restriction of Lebesgue measure to the Borel
subsets of $\mathbf R$.
- We define the symmetric difference of two subsets $E$ and
$F$ to be $E\Delta F:=(E\setminus F)\cup (F\setminus E)$. In this
problem, you may use without proof that every open subset of
$\mathbf R$ is a countable disjoint union of open intervals.
Suppose that $E\subset \mathbf R$ is a set of finite Lebesgue
measure. Let $\epsilon>0$. Show that there is finite disjoint
union $F$ of open intervals such that $m(E\Delta F)<\epsilon$.
- Let $(X,\mathcal M,\mu)$ be a measure space and let $(X,\mathcal
M_0,\mu_0)$ be its
completion.
- If $f:X\to\mathbf C$ is $\mu_0$-measurable, show
that there is a $\mu$-measurable function $g:X\to\mathbf
C$ such that $f=g$ for $\mu_0$-almost all $x$. (Hint:
show that it suffices to assume that $f$ is a
$\mu_0$-measurable simple function.)
- Further observe that there is a $\mu$-null set $N$
such that $f(x)=g(x)$ if $x\notin N$.
- What does this result say about Lebesgue measurable
functions on $\mathbf R$ and Borel functions? (See the
last part of the question #37.)
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Monday, October 26:
- Homework Solutions: Here
are selected solutions for
the homework. (Last modified December 31, 1969)
- Work:
- Show that a Hahn decomposition for a real-valued measure
$\nu$ is unique up to null sets as claimed in lecture. Also
verify that the singular measures in the Jordan decomposition
of a real-valued measure $\nu$
are unique.
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Wednesday, October 28:
- Work:
- Suppose that $(X,\mathcal M,\mu)$ is a measure space and
that $f:X\to[0,\infty]$ is measurable. Let $\nu$ be the
measure on $(X,\mathcal M)$ defined by $$\nu(E)=\int_E
f(x)\,d\mu(x)\quad\text{for all $E\in\mathcal M$.}$$ Show that
if $g:X\to\mathbf C$ is measurable, then
$g\in \mathcal L^1(X,\nu)$ if and only if $fg\in \mathcal
L^1(X,\mu)$ and that
$$\int_X g(x)\,d\nu(x)=\int_X g(x)f(x)\,d\mu(x). $$ (Hint:
start with the corresponding result from Lecture 14.)
- Let $\mu$, $\nu$, and $\lambda$ be $\sigma$-finite measures
on $(X,\mathcal M)$. We'll denote the Radon-Nikodym derivative of $\nu$
by $\mu$ by $\displaystyle{
\frac{d\nu}{d\mu}}$.
- Show that $\displaystyle{
\frac{d\nu}{d\mu}}$ is determined $\mu$-almost everywhere.
- Suppose that $\nu\ll\mu\ll\lambda$. Show that
$\displaystyle{{\frac{d\nu}{d\lambda}}=
\frac{d\nu}{d\mu}\frac{d\mu}{ d\lambda}}$. Of
course, "$=$" means "equal almost everywhere $[\lambda]$."
- Suppose that $\mu\ll\nu$ and $\nu\ll\mu$ (we say the
$\mu$ and $\nu$ are equivalent and write $\nu\approx\mu$). Show
that $\displaystyle{\frac{d\mu}{d\nu}=
\left[\frac{d\nu}{d\mu}\right]^{-1}}$. Again
"$=$" means "equal almost everywhere $[\mu]$ (or
$[\nu]$)".
- Show that the $\sigma$-finite hypothesis is necessary in the
Radon-Nikodym theorem. (Hint: let $\nu$ be Lebesgue measure on
$[0,1]$ and let $\mu$ be counting measure (restricted to the
Lebesgue measurable sets in $[0,1]$).)
- Suppose that $\rho$ is a premeasure on an algebra $\mathcal
A$ of sets in $X$.
- Show that $$\rho^*(E)=\inf\{\sum_{k=1}^\infty
\rho(A_k):\text{each $A_k\in\mathcal A$ and $E\subset \bigcup
A_k$}\}$$ is an outer measure on $X$.
- Show that $\rho^*(A)=\rho(A)$ for all $A\in\mathcal A$.
- Show that each $A\in \mathcal A$ is $\rho^*$-measurable.
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Friday, October 30:
- Cantor-Lebesgue Function: At one point, several of
you expressed interest in the relationship between the Cantor
set and ternary expansions. This leads to
the Cantor-Lebesgue function which has some interesting
properties. It can also be used to give
another proof that there are Borel
subsets of $\mathbf R$ that are not Lebesgue measurable.
Follow this link for a discussion
of this. Doing so is entirely optional and this material is not
officially part of the course.
- Homework:Problems 36-45 will be due Friday,
November 6 via gradescope.
- Work:
- Show that the conlusion of
Egoroff's Theorem can fail if $\mu(X)=\infty$,
but is still valid if "$\mu(X)<\infty$" is replaced by
"$|f_n(x)|\le g(x)$ with $g\in\mathcal L^1(\mu)$". (For the
statement of Egoroff's Theorem and the proof in the finite
measure case, see Lecture 18.)
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Monbday, November 2:
- Homework:Problems 36-45 will be due Friday,
November 6 via gradescope.
- Work:
- Let $\mathcal B(\mathbf R)$ be the Borel $\sigma$-algebra
in $\mathbf R$ and $\mathcal B(\mathbf R^2)$ the Borel
$\sigma$-algebra in $\mathbf R^2$. Show that $\mathcal
B(\mathbf R)\otimes \mathcal B(\mathbf R)=\mathcal B(\mathbf
R^2)$. (You may use the observation that every open set in
$\mathbf R^2$ is a countable union of open rectangles.)
- Let $(X,\mathcal M,\mu)$ and $(Y,\mathcal N,\nu)$
be complete $\sigma$-finite
measure spaces. Let $(X\times Y,\mathcal
L,\lambda)$ be the completion of $(X\times Y,\mathcal M\otimes
\mathcal
N,\mu{\times}\nu)$.
- Suppose that $E\in \mathcal M\otimes \mathcal N$ and
$\mu{\times}\nu(E)=0$. Show that $\mu(E^y)=0=\nu(E_x)$ for
$\mu$-almost all $x$ and $\nu$-almost all $y$.
- Suppose that $f$ is $\mathcal L$-measurable and that
$f(x,y)=0$ for $\lambda$-almost all $(x,y)$. Show that
there is a $\mu$-null set $M$ and a $\nu$-null set $M$ such
that for all $x\notin M$ and $y\notin N$, $f_x$ and $f^y$
are integrable and that
$$\int_X f^y\,d\mu =0=\int_Y f_x\,d\nu.$$
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Friday, November 6:
- Let $\mathcal B$ be the Borel $\sigma$-algebra in $[0,1]$. Let $m$
be Lebesgue measure on $([0,1],\mathcal B)$ and let $\nu$ be
counting measure on $([0,1],\mathcal B)$. Let
$D=\{\,(x,x)\in [0,1]\times[0,1]:x\in [0,1]\,\}$ be the diagonal.
Compare the integrals
\begin{equation}
\label{eq:40}
\int_{[0,1]\times[0,1]}1_D\,d (m{\times}\nu),\quad
\int_{[0,1]}\int_{[0,1]}1_D\,dm\,d\nu,\quad\text{and}
\quad\int_{[0,1]}\int_{[0,1]}1_D\,d\nu\,dm,
\end{equation}
and comment on the relationship to the Fubini-Tonelli Theorems.
(You will have to resort to the definition of $m\times\nu$ to
compute the first integral.)
- Let $(X,\mathcal M,\mu)$ and $(Y,\mathcal M,\nu)$ both be counting
measure $(\mathbf N,\mathcal P(\mathbf N),\nu)$. Define
$f:X\times Y\to \mathbf R$ by
\begin{equation}
\label{eq:46}
f(m,n)=
\begin{cases}
\phantom{-}1&\text{if $m=n$,}\\-1&\text{if $m=n+1$,
and}\\\phantom{-}0&\text{otherwise.}
\end{cases}
\end{equation}
Observe that $f$ is not integrable and that the two iterated integrals
$\int_{Y}\int_{X}f\,d\mu\,d\nu$ and $\int_{X}\int_{Y}f\,d\nu\,d\mu$
are not equal. What is the moral here?
- Let $(X,\mathcal M,\mu)$ be a measusre space.
- Show that $\|\cdot\|_\infty$ is a norm on
$L^\infty(X)$.
- Show that $f_n\to f$ in $L^\infty(X)$ if and only if there
is a null set $E\in\mathcal M$ such that $f_n\to f$ uniformly
on $X\setminus E$.
- Show that $L^\infty(X)$ is a Banach space.
- Show that (measurable) simple functions are dense in
$L^\infty(X)$. (Hint: re-read our result on MNNSFs from
Lecture 12.)
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!-- Friday's assignment -->
Monday, November 9:
- Homework Solutions: Here
are selected solutions for
the homework. (Last modified December 31, 1969)
- Work:
- [OPTIONAL -- DO NOT TURN IN] In the notation of our
technical proposition from Lecture 25, show that if $\mu$ is
semifinite, $q<\infty$, and $M_q(g)<\infty$, then
$\{\,x:|g(x)|>\epsilon\,\}$ has finite measure for all
$\epsilon>0$. Hence $S_g$ is $\sigma$-finite as required in
the proof.
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!-- Friday's assignment -->
Wednesday, November 11:
- Let $(X,\mathcal M,\mu)$ be a measures space with
$\mu(X)<\infty$. If $1\le r < s\le \infty$, show that $\mathcal
L^s(X)\subset \mathcal L^r(X)$. (Hint: apply Holder to
$|f|^r=1\cdot |f|^r$ where $1$ denotes the constant
function.)
- Show that if $1\le r< s \le \infty$, then $\ell^r\subset
\ell^s$.
- Let $(X,\mathcal M,\mu)$ be a measures space. Suppose that
$1\le r< t < s\le \infty$. Show
that $\mathcal L^r(X)\cap \mathcal
L^s(X)\subset \mathcal L^t(X)$.
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!-- Friday's assignment -->
Friday, November 20:
- Homework Solutions: Here
are selected solutions for
the homework. (Last modified December 31, 1969)
|
