Tagged Questions
1
vote
0answers
36 views
Measurability is true?
Can you please help me solve this on measurablilty? My TA did not go over this in measurability. He said we are not going over this but you can do this if you want. Can someone please explain to me. ...
0
votes
0answers
25 views
Measurablilty with infinity
Can someone please show me how to solve this wth measurability? My TA did not show us this yet but I was curious how to solve it.
I know that $1<x<n$ for ${\frac{1}{x^2}}$ and the $sup$ $=$ lim ...
1
vote
1answer
55 views
Show that a function is continuous
Let K be bounded and continuous and bounded on $\mathbb{R}^{n}$ and let $f$ be Lebesgue integrable on $\mathbb{R}^{n}$.
Show that the function $g$ defined on $\mathbb{R}$ by
$g(t) = ...
0
votes
2answers
36 views
Riemann integral and Lebesgue integral
$f:R\rightarrow [0,\infty)$ is a Lebesgue-integrable function. Show that
$$
\int_R f \ d m=\int_0^\infty m(\{f\geq t\})\ dt
$$
where $m$ is Lebesgue measure.
I know the question may be a little dump.
...
3
votes
2answers
59 views
Proof of Egoroff's Theorem
Let $\{f_n \}$ be a sequence of measurable functions, $f_n \to f$ $\mu$-a.e. on a measurable set $E$, $\mu(E) < \infty$. Let $\epsilon>0$ be given. Then $\forall \space n \in \mathbb{N} \space ...
1
vote
1answer
47 views
Composition of Lebesgue measurable function $f$, with a continuous function $g$ having a certain property, is Lebesgue measurable
Suppose that $f$ is Lebesgue measurable and $g$ is real valued, continuous, and has the property that for any null set $N$, $g^{-1} (N)$ is measurable. Then $f \circ g$ is also Lebesgue measurable. ...
1
vote
1answer
51 views
A question about “nice” functions
Let $f:\mathbb R \rightarrow \mathbb R$ be a function such that $\lambda(I)=\lambda(f(I))$ for each interval $I \subseteq \mathbb R$. ($\lambda$ is Lebesgue measure here.) Let us call such functions ...
1
vote
1answer
51 views
Integral over null set is zero but integral of Dirac delta function is 1
We know integral of any function over a null set is zero.
But for Dirac delta function ($\delta=+\infty$ iff $x=0$ otherwise $\delta=0$)
$$
\int_{-\infty}^{+\infty}\delta =\int_0^0\delta =1.
$$
Is it ...
1
vote
1answer
25 views
$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$ for $f\in L^{p}$, $p \in [1,\infty)$
For $f\in L^{p}$, $p \in [1,\infty)$
we want to prove:
$$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0$$
I'm not sure whether we can exchange the limit and the integral, cuz I cannot find ...
1
vote
0answers
22 views
Lebesgue measure of set $M = \{ [x,y] \in \mathbb{R}^2; 2 < x + y < 3; x < y < 3x \}$?
although we can do this by splitting the area four ways and computing four integrals, my book suggests that I try the substitution $ u = x + y$ and $ v = \frac{y}{x}$.
I expressed $x$ and $y$ in ...
3
votes
1answer
80 views
If $f$ is a bounded measurable function $\Longrightarrow$ there is a sequence of step functions such that $s_n \longrightarrow f \; a.e$?
If $f:[0,1]\longrightarrow\mathbb{R}$ is a bounded measurable function $\Longrightarrow$ there is a sequence of step functions $\displaystyle s_n=\sum_{j=1}^{p} c_j \cdot \chi _{I_j}$ such that $s_n ...
0
votes
1answer
34 views
Show that E is measurable?
Suppose $E_1= [1, 1 \frac12] , E_2 = (2, 2\frac14), E_3 = [3, 3\frac18], E_4 = (4 , 4 \frac{1}{16}) , \dots , E= \bigcup_{n=1}^{\infty}E_n
$
i) Show $E$ is measurable
ii) Compute $m(E)$
Here is ...
0
votes
0answers
62 views
Let $g$ be a bounded measurable function on $[0,1]$.
Let $g$ be a bounded measurable function on $[0,1]$.
For each $n$
Let $\displaystyle I_j=j\cdot \frac{1}{2^n}+[0,\frac{1}{2^n}] $ , $j=0,1\cdots ,2^n-1$ , a partition of $[0,1]$ by bisections
...
2
votes
1answer
51 views
Riemann integral with intervals?
Let $f(x) = \begin{cases} 3 && 0 \leq x \leq 1 \\ 0 && 1 \leq x \leq 2 \end{cases}$
Compute $\displaystyle \ \ \int_0^2 f(x)dx\,\,\,$.
You can use the definition of Riemann integral ...
1
vote
1answer
36 views
Counterexample to upper continuity
Let $M$ be a $\sigma$-algebra of subsets of a set $X$ and let $\mu:M\rightarrow[0,\infty)$ be a finitely additive set function. I'm trying to decide if it's automatically true that for all ascending ...
0
votes
1answer
56 views
E measurable with m(E) < $\infty$?
Suppose that $E$ is measurable with $m(E)$ $<$ $\infty$.
ii) Show that $\displaystyle \ \ \int_E 2f\,\,\,$ $=$ $2$$\displaystyle \ \ \int_E f\,\,\,$ if $f$ is bounded and measurable.
I told my ...
0
votes
1answer
78 views
If $f :\mathbb{R}\to\mathbb{R}$ is measurable, then $E = \{x: f(x) \geq 3\}$ is measurable
Prove: Suppose $f : \mathbb{R}\to\mathbb{R}$ where $f$ is measurable and $E = \{x: f(x) \geq 3\}$. Show $E$ is measurable.
I saw this statement while reading in a paper and thought this might ...
-1
votes
0answers
24 views
Lebesgue inner measure
the definition of inner measure: m ∗ (A)=sup{m(S):S∈M,S⊆A} I need to prove:
1) If inner measure=outer measure then A is measurable set
2) m*(AUC)+m*(A n C)>m*(A)+ m* (C)
3)m*(UA)> sum (m*(A)) for ...
0
votes
0answers
31 views
question about essential supremum
Consider $u : \Omega \rightarrow \mathbb{R}$ a measurable, nonnegative and bounded function. With $\Omega \subset \mathbb{R}^n$ bounded and open .
Is true that $\mathrm{ess } \inf \ u = \inf \ ...
2
votes
0answers
26 views
Lebesgue Integral Rudin Problem [duplicate]
Suppose {$n_k$} is an increasing sequence of positive integers and E is the set of all x$\in$($-\pi, \pi$) at which {sin$n_k x$} converges. Prove that $m(E)=0$.
Hint: For every A $\subset$ E, ...
2
votes
1answer
59 views
About measure theoretic interior and boundary
Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery.
I just want to clarify whether these definitions of measure theoretic interior
and boundary are correct. Given ...
0
votes
1answer
39 views
Showing that a union of the subsets of two $\sigma$-algebras is a $\sigma$-algebra.
I got back an assignment for a first course in analysis and I have made a very basic error, and I'm having a lot of trouble pinpointing exactly what piece of information I'm missing.
You have two ...
2
votes
2answers
37 views
Measure Not Preserving Homeomorphism
Are there homeomorphisms that do not preserve measure (Let us consider Lebesgue Measure)? I ask this because, I currently cannot think one but while googling, I get Measure Preserving Homeomorphisms. ...
2
votes
1answer
45 views
dominated convergence theorem
I am studying the proof of a theorem and in a part of the proof I have the following situation:
Let $u : \Omega \rightarrow R$ a nonnegative measurable function, with $\Omega$ open and bounded. ...
6
votes
1answer
121 views
Working on a generalized Cantor set, with Lebesgue measure, and a certain inequality.
Let's consider the interval $[0,1]$ in the same way that we constructed the Cantor set, we can use the same idea, but instead of removing in the step $n$ middle open intervals of length ...
2
votes
1answer
132 views
Is $f$ necessarily measurable?
(1) Suppose a function $f$ has a [Lebesgue] measurable domain and is continuous except at a finite number of points. Is $f$ is necessarily [Lebesgue] measurable?
Comments For (1), If $f$ is defined ...
1
vote
0answers
53 views
How to prove this set has volume 0?
Let $A \subset \mathbb R^n$ be an open set and the function $g: A\to \mathbb R^n$ be a $C^1$ function. Let
$$B=\{x\in A: Jg(x)=0\}$$ ($Jg$ means the Jacobian of $g$), then $g(B)$ is of volume zero. ...
2
votes
1answer
62 views
What is the difference between the spaces $L^1$($\mu$) and $L^1$(d$\mu$)? And is one a subset of the other?
What is the difference between the spaces $L^1$($\mu$) and $L^1$(d$\mu$) ? And is one a subset of the other?
$\mu$ is the Lebesgue measure.
1
vote
1answer
97 views
Determining the Lipschitz constant
Determine the corresponding Lipschitz constant of $f(t,y(t))=e^{(t-y)/2}$, where $D=\{(t,y) : 0\leq t \leq 1,-\infty<y<+\infty\}$.
0
votes
0answers
70 views
Show $f$ is in $L^1$ (d$\mu$) space and $_X\int f $ d$\mu=\lim_{n\to \infty}\int_X f_n d\mu$
Suppose $f$ is in $L^1$($\mu$). Prove that for each $\epsilon > 0$ , there exists a $\delta > 0$ so that the $\int |f|\mathrm d\mu$ < $\epsilon$ over the set $E$ whenever $\mu(E) < ...
1
vote
1answer
71 views
Range of function measurable?
Consider a function $f:\mathbb{R}\to\mathbb{R}$.
If $f$ is Borel measurable then is the range $f(\mathbb{R})$ Borel measurable (or maybe Lebesgue measurable).
Likewise if $f$ is Lebesgue measurable ...
0
votes
1answer
34 views
A clarification about BV functions.
From definition, a locally integrable $ u \in BV(\Omega) $ if its distribution derivative is given by a signed Radon measure. That is there exists $ \mu $ such that for any $ \phi \in ...
0
votes
0answers
31 views
Arbitrarily small open set [duplicate]
Let $E \subset \Re$ be measurable with $\mu(E) > 0$. Show that there
is an $\epsilon > 0$ such that $(-\epsilon, \epsilon) \subset \{x-y \mid x \in E,\, y \in E\}$.
Solution:
...
1
vote
1answer
19 views
topology in extended real line
I am having trouble with a simple question :
Consider $\bar{R}$ the extended real line and $ 0 < q < \infty$.
Let $x_n $a sequence in $\bar{R}$ with $x_n \geq 0, \forall \ n $. Suppose that ...
0
votes
0answers
55 views
Continuity of normed linear spaces
Let $A: X \rightarrow Y$ be linear and $X$ and $Y$ are normed linear
spaces. Show that $A$ is continuous if and only if $A$ is bounded.
Solution:
$A$ is bounded if and only if there is a $M ...
1
vote
1answer
43 views
$f$ maps measurable sets to measurable sets
Show that if $f:\mathbb{R} \rightarrow \mathbb{R}$ is absolutely continuous, then $f$ maps measurable sets to measurable sets.
Any ideas on how to do this?
1
vote
1answer
28 views
Measure and the intersection with an open interval
Let $E \subset \Re$ be measurable with $\mu(E) > 0$. Show that for
every $0 < \epsilon < 1$ there is an open interval $I$ such that
$\mu(E \bigcap I) > (1-\epsilon)\mu(I)$
Let $0 ...
3
votes
1answer
45 views
Every set $E \subseteq \mathbb{R}$ of positive measure is the disjoint union of two sets $E = B \cup C$ such that $\mu(B) = \frac12\mu(E) = \mu(C)$
Let $E \subset \Re$ be measurable with $\mu(E) > 0$. Show that there
are disjoint sets $B$ and $C$ such that $E=B \cup C$ and $\mu(B) =
\mu(C) = \frac12\mu(E)$.
Solution:
Assume $\mu(E) = ...
0
votes
0answers
38 views
Open set in measure greater than 0 [duplicate]
Let $E \subset \Re$ be measurable with $\mu(E) > 0$. Show that there
is an $\epsilon > 0$ such that $(-\epsilon, \epsilon) \subset \{x-y \mid x \in E,\, y \in E\}$.
Solution:
...
1
vote
1answer
39 views
Vitali covering problem
Let $ R $ denote the set of rational numbers in $ [0,1] $ and for each $ r∈R $, let $ V_r = \{ [r,r+ 1/k] : k=1,2,3,\ldots\} $. Put $V = ⋃_r V_r $.
Show that for every $ ε > 0 $ there exists a ...
1
vote
0answers
25 views
Not every $1$-nullset is an $s$-nullset, for $0 < s < 1$
For any positive real number $s$, define an $s$-nullset as a subset $A$ of the real line such that, for any $\epsilon > 0$, there exists a sequence of intervals $\{I_n\}_{n=1}^{\infty}$ having the ...
1
vote
0answers
22 views
Jordan measure on the unit interval
It is known that Jordan measure is finitely additive measure and the set of all Jordan measurable forms a Boolean algebra $\mathcal{B}$. My question is ?
Show that the Jordan measure on $[0,1]$ ...
4
votes
1answer
60 views
Discontinuous Almost-Everywhere/ Unbounded in $L^{1}(\mathbb{R})$
Let $L^{1}(\mathbb{R})$ be defined as usual, with the equivalence relation : $f \approx g$ if and only if $f(x) = g(x)$ almost everywhere.
Is there a class in $L^{1}(\mathbb{R})$ such that every ...
0
votes
2answers
21 views
Prove $\exists\;E$ such that $\mu(E)=0$ and $\alpha=\sup\limits_{x\in X\backslash E} |f(x)|$
$(X,\mathscr{M},\mu)$ is a measurable space and $f$ is a measurable function on $X$. Denote by $$\alpha=\inf\{\sup\limits_{x\in X\backslash E} |f(x)| : E\in\mathscr{M}, \mu(E)=0\}.$$
Note that the ...
3
votes
1answer
38 views
Prove that $\lim\limits_{n\to\infty} \int\limits_X f_nd\mu=\int\limits_X fd\mu$
$(X,\mathscr{M},\mu)$ is a measurable space and $f_n,g_n,f,g$ are all measurable functions defined on $X$. The following conditions are satisfied:
(i) $f_n\to f\;\&\;g_n\to g$ almost everywhere ...
0
votes
0answers
49 views
Creating non-measurable functions from Cantor function
I encountered the following problem in which I don't understand the construction it is proposing:
The Cantor d.f. $F$ is a good building block of "pathological"
examples. For example, let ...
0
votes
1answer
19 views
Outer Measure of a Finite Covering of the Rationals on $[0, 1]$
I'm studying for my Real Analysis final and came upon an old question on outer measure that I'm pretty sure I'm doing wrong.
If $B$ is the set containing the rationals on $[0, 1]$, and ...
1
vote
2answers
51 views
Question in Lebesgue integrable functions.
Suppose $g$ be a measurable function satisfying: $∀$ $σ∈[c,d]$ , there exists $δ>0$ such that $∫_E|g| <∞$ where $E=[σ-δ, σ+δ]$. Prove that $g$ is Lebesgue integrable on $[c, d]$.
2
votes
1answer
35 views
about well-defined integral kernel
Let $\phi:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ measurable function such that
$$ \int_{\mathbb{R}^n}|\phi(x,y)|\ dx \leq M\ , \quad \int_{\mathbb{R}^n}|\phi(x,y)|\ dy \leq M\,.$$
Let $f\in ...
0
votes
0answers
32 views
Combining convergence in probability and the means of the positive sequence of r.v. implies convergence in L 1
Let $\{X_n\}$ be a collection of positive random variable with $X_n \rightarrow X$ in probability. Prove that if $E(X_n) \rightarrow E(X)$, then $X_n \rightarrow X$ in $L^1$.
My partial answer:
Let ...
