Tagged Questions

1
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0answers
27 views

Show derivative of integral equals integral of partial derivative if M[0,1]-measurable

I am trying to determine a method of approaching the following: Suppose that $f:[0,1] \times (0,1)$ $\rightarrow$ $\mathbb{R}$ is such that, for each $y \in (0,1)$, the function $f^{[y]}(x) = f(x,y)$ ...
0
votes
0answers
67 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) < ...
2
votes
1answer
50 views

Interchange differential operator with Lebesgue integral.

Under what condition am I able to interchange a differential operator with an integral? More precisely, given a function $f:\Omega\times U\to\Bbb R$ from a measure space $(\Omega,\mathscr A,\mu)$ and ...
2
votes
1answer
34 views

Abstract integral - Borel measures - $L^p$ spaces

Let $(X,\mu,M)$ be a finite measure space. Suppose $T\colon X \to X$ is measurable and $\mu(T^{-1}E) = 0$ whenever $E \in M$ and $\mu(E)=0$. Prove that these exists $h \in L^1(\mu)$ such that $h ...
2
votes
1answer
108 views

Riemann-Stieltjes integrability criterion

I am currently reading through chapter 11 of Rudin's Principles of Mathematical Analysis, and I'm trying to solve problem 7: Find a necessary and sufficient condition that $f \in \mathfrak R(\alpha)$ ...
0
votes
0answers
28 views

Question about probability measures on the real line [closed]

http://www2.imperial.ac.uk/~boz/M34P6/P11_2.pdf Dear comrades. I am struggling with Ex 1.4(i) on here. I think that I have proved that the measures $\mu$, $\nu$, $\lambda$ are all equivalent, in the ...
-6
votes
1answer
112 views

Convergency of the functions $ϕ_nf_n$ converge to $ϕf$ in $L_1(μ)$

Suppose that a sequence of $μ$-integrable functions $f_n$ converges to $f$ in $L_1(μ)$ and a sequence of $μ$-measurable functions $ϕ_n$ converges to$ϕ$ $μ$-a.e. and is uniformly bounded. Show that the ...
1
vote
1answer
62 views

Question about integration (related to uniform integrability)

Consider a probability space $( \Omega, \Sigma, \mu) $ (we could also consider a general measure space). Suppose $f: \Omega -> \mathbb{R}$ is integrable. Does this mean that $ \int |f| \chi(|f| ...
1
vote
1answer
32 views

Quotient of two $L^1$ functions

Suppose that we have two $L^1(\mathbb{R})$ functions, $f$ and $g$, where $g$ is non-zero almost everywhere. Can we conclude that their quotient $\frac{f}{g}$ is finite almost everywhere?
6
votes
1answer
145 views

How to find a measurable but not integrable function or a positive integrable function?

For an arbitrary interval $I$, how can we find a positive on $I$ integrable function? And how does one construct a measurable but not integrable function. If not all measurable functions are ...
0
votes
0answers
33 views

Is it possible to show that if $-f , f \in L^{+}(I)$ then $f\in R(I)$

Is it possible to show that if $-f , f \in L^{+}(I)$ then $f\in R(I)$? We can rewrite the integrals of $-f$ and $f$ on $I=[a,b]$ as: $$\int_{a}^{b} fdx = \lim_{n\rightarrow ...
2
votes
2answers
179 views

Does Riemann integrable imply Lebesgue integrable?

Suppose a definite integral exists in the Riemann sense. Does that mean the integral exists as a Lebesgue integral, and do we get the same result either way?
1
vote
1answer
75 views

Construction of a sequence of simple functions converging pointwise to a given function

Q1: How to construct a sequence of $\{f_n\}$ of simple function for a function $f$ such that $f_n\to f$ converges pointwise? Q2: If $f$ is measurable is $f_n$ also measurable for each $n$? Q2 is by ...
0
votes
0answers
61 views

Question about an integration method in Analysis

I have a question about an integration method widely used in Analysis, namely the fact that $$ \int_{B(x_0,R)} { \hspace{-20pt} f(x)\,{\rm d} x } = \int_0^R { \hspace{-5pt} ...
3
votes
0answers
209 views

Riemann integral vs Lebesgue integral

Let $f$ be analytic on a domain $\Omega$ of the complex plane, such that the closed disc $\overline{D(0,R)}$ is contained in $\Omega$. What is the difference between $$ \int_{D(0,R)}|f(w)|dm(w)$$ and ...
1
vote
0answers
45 views

Lebesgue-Stieltjes integral as a generalized function

Given some convex function $f(x)$, $x >0$ we can define a distribution $F \in \mathcal{D}'(0,\infty)$ using Lebesgue-Stieltjes integral $$ \langle F, \varphi \rangle ...
3
votes
2answers
99 views

Showing that $\lim_{x\rightarrow 0} \frac{1}{x}\int_0^x |\sin(1/y)| \mathrm{d} y \not=0$

How to show that: $$\lim_{x\rightarrow 0} \frac{1}{x}\int_0^x |\sin(1/y)| \mathrm{d} y \not=0$$ It seems like a easy example of illustrating 0 is not in the Lebesgue set of $g(x)$ where ...
0
votes
4answers
96 views

Proof that the open Ball could not be written as a finite union of intervals

An intervall in $\mathbb{R}^n$ is a set of points $x = (x_1, \ldots, x_n)$ such that $$ a_i < x_i < b_i \qquad (i = 1, \ldots, n) $$ and where $<$ could also be replaced by $\le$. If $A$ is ...
1
vote
2answers
87 views

Equicontinuous, differentiable continuous problem

Assume that each of {$f_n : [0, 1] \rightarrow R$} is continuously differentiable I know that if {$f_n'$} is uniformly bounded, {$f_n$} is equicontinuous. However, the converse is NOT true. I want ...