For question about integration, where the theory is based on measures. So it's almost always used together with the tag [measure-theory], and its aim is to specify questions about integral, not only properties of the measure.
0
votes
2answers
44 views
Proving that $L^p \subset L^q$ when $1 \le q \le p$ [duplicate]
Let $(E,\mathcal{F},\mu)$ be a measure space such that $\mu(E)=1$ and let $L^p=L^p(E, \mathcal{F},\mu)$. Prove that
$L^p \subset L^q\text{ if } 1 \le q \le p$.
I let $f \in L^p$. Then $(\int_E ...
3
votes
0answers
54 views
How to prove $\mathcal{L}^2[(0,1)]$ is a Hilbert Space
Let $\mathcal{L}^2[(0,1)]$
denote the set of $\mathbb{C}$-valued square integrable functions on the interval [0,1].
Prove that $\mathcal{L}^2[(0,1)]$ forms a Hilbert Space.
I believe that I can ...
0
votes
0answers
47 views
let $f(x,y)=\frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}$.
let $f(x,y)=\frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}$.
Show that
$\int_{0}^{1} dx \int_{0}^{1} f(x,y) dy=\frac{\pi}{4}$
$\int_{0}^{1} dy \int_{0}^{1} f(x,y) dx=-\frac{\pi}{4}$
0
votes
0answers
29 views
Lebesgue integrable functions
Suppose
$$f(x) = \sum_{n=0}^{\infty} \frac{1}{2^n} \phi(2^n x)$$ on the closed interval $[0,1]$ where $\phi$ is given by
$$\phi(x) = \begin{cases} x, & \mbox{if } x \in \left[0,\frac12\right] \\ ...
1
vote
0answers
34 views
Show derivative of integral equals integral of partial derivative if M[0,1]-measurable
I am trying to determine a method of approaching the following:
Suppose that $f:[0,1] \times (0,1)$ $\rightarrow$ $\mathbb{R}$ is such that, for each $y \in (0,1)$, the function $f^{[y]}(x) = f(x,y)$ ...
0
votes
1answer
28 views
How to show this dual space pairing is measurable
If $f \in L^2(0,T;H^{-1})$ and $g \in L^2(0,T;H^1)$, how to show that $\langle f(t), g(t) \rangle_{H^{-1},H^1}$ is measurable over [0,T]?
If it's measurable, it's clearly integrable. But how to show ...
1
vote
1answer
25 views
Sufficient conditions for $z \to \int_\mathbb{R} h(z,x) \,d\mu(x)$ to be analytic
Let $\mu$ be a measure on $\mathbb{R}$ and $G$ an open subset of $\mathbb{C}$. Every function $h \,:\, G\times\mathbb{R} \to \mathbb{C}$ then gives rise to a function $$
F_h \,:\, G \to \mathbb{C} ...
0
votes
0answers
74 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
65 views
Limit of Lebesgue integral in $L_1([-1,1],m)$
Let $([-1,1],\mathcal{M},m)$ be a measurable space in $[-1,1]$ where $m$ is the Lebesgue measure in $\mathbb{R}$ restricted to $[-1,1]$, and $\mathcal{M}$ is the set of $m^*$-measurable subsets of ...
1
vote
1answer
42 views
Limit of Lebesgue integrals in $L_1(\mathbb{R},m)$
Let $g\in L_1(\mathbb{R},m)$ bounded function where $m$ is the Lebesgue measure in $\mathbb{R}$. If
$$\lim_{x\to\pm\infty} g(x)=0,$$
show that for all functions $f\in L_1(\mathbb{R},m)$ we have:
...
4
votes
2answers
89 views
Infinite shots fired in a lattice forest
A hunter is standing in the center of an infinite 2D forest. There are point trees at all the integer lattice points. The hunter fires a gun with a bullet of zero width in a random direction. He ...
0
votes
1answer
41 views
Consequence of Lebesgue Monotone Convergence Theorem
If $a_{ij}\geq0$ if $i,j\in \mathbb{N}$, show using LMCT that,
$$\sum_{i=1}^\infty\sum_{j=1}^\infty a_{ij}=\sum_{j=1}^\infty\sum_{i=1}^\infty a_{ij}.$$
Attempt: Simply I can't realize how to use ...
5
votes
1answer
94 views
Homogenous measure on the positive real halfline
Define a measure $\mu\not=0$ on positive real number $\Bbb R_{>0}$ such that for any measurable set $E\subset\Bbb R_{>0}$ and $a\in \Bbb R_{>0} $, we have $\mu(aE)= \mu(E)$, where ...
1
vote
1answer
33 views
Lebesgue integral is linear in simple functions.
Let $(X,\mathcal{M},\mu)$ be a measure space. Let $s,t: X\to[0,\infty)$ two simple functions. If $E\in\mathcal{M}$, show that,
$$\int_E (s+t)\,d\mu=\int_E s\,d\mu+\int_E t\,d\mu$$
Attempt:
...
1
vote
1answer
35 views
Finding lebesgue integral of piecewise function
Consider the function $f(x)=\begin{cases}
x & x\in \mathbb{Q}\cap [0,1] \\
-x & x\in [0,1]-\mathbb{Q} \\
\end{cases}$
How do you go about computing the following Lebesgue ...
-1
votes
2answers
54 views
prove $ e^{bt} \ \int_0^t \ f(s) ds=\int_0^t \ ( e^{-bs} \ f(s)-be^{-bs}\int_0^s\ f(u)\ du) \ ds $ [duplicate]
please How can I prove that
$$
e^{-bt} \ \int_0^t \ f(s) ds=\int_0^t \ ( e^{-bs} \ f(s)-be^{-bs}\int_0^s\ f(u)\ du) \ ds
$$
f non-negative measurable function
I would appreciate it enormously if ...
0
votes
1answer
47 views
prove that integral
prove that
$$-\int_0^t \ sgn(f({s})) \ d{s}=\int_0^t \ sgn(-f({s}))\ d{s}+2\int_0^t 1_{f({s}) =0}d{s}$$
with
$$ sgn(x) := \begin{cases}
-1 & \text{if } x =< 0, \\
1 & \text{if } x > ...
1
vote
0answers
29 views
The tightest bound on an integral
Consider a polynomial $p(x)$ where $p(x)>0$ for $x\in(0,1)$ and $p(0)=0$. Let $s(x)$ be an increasing analytic function such that $s(0)=0$ and $s(1)=1$. I am interested to bound the following ...
2
votes
1answer
29 views
measure of the image of the the unit open disc by a holomorphic map
I found the following interesting exercice in a textbook:
Let $f$: $\Bbb E \to \mathbb{C}$ be a holomorphic and injective map ('Schlicht function'), where $\Bbb E=\{z \in \Bbb C:|z|<1\}$. ...
0
votes
1answer
34 views
Lebesgue integrable function and square-integrable functions?
I heard that lebesgue-integrable function is square-integrable function, and vice versa. Why is this the case? (in my textbook) On the definition of Lebesgue integration, it only defines that ...
-2
votes
1answer
90 views
Showing that $\{f_n \}$ converges to $f$ is equivalent to $\lim_{n\to \infty} \int_X \frac{|f_n(x)-f(x)|}{1+|f_n(x)-f(x)|}d\mu(x)=0$
Let $(X,\mathcal{M},\mu)$ be a measure space. We say that $\{f_n\}$ converges to $f$ in measure if, for any $\epsilon>0$
$$ \lim\limits_{n \to \infty} \mu\Big(\{x\in X: |f_n(x)-f(x)| \ge ...
1
vote
1answer
41 views
Prove $ F(x)=\int_0^{\infty}\frac{f(y)}{x+y}dy $ is continuous on $(0,\infty)$ and differentiable, and have $\lim\limits_{x\to \infty} F(x)=0$.
$f$ is integrable on $[0,\infty)$, and $\int_0^{\infty} |f(y)|dy < \infty$.
Prove:
Then $ F(x)=\int_0^{\infty}\frac{f(y)}{x+y}dy $ is continuous on $(0,\infty)$ and differentiable, and have ...
1
vote
1answer
30 views
Confusing application of DCT
I need to apply DCT to bring the limit inside the abslute value...
$\displaystyle \lim_{\tau \rightarrow 0} \left\lbrace \sum_{z \in \mathcal{E}} \mid a_z \mid ^2 \left(\int_{B_{R}} ...
1
vote
0answers
34 views
continuous on Lebesgue space
$f\in \mathcal{L}^{\infty}(R), t>0,x \in R, B(x,t)=\{ y\in R: |x-y|<t \}$, and $u(x,t)$ is defined as:
$$ u(x,t)= \frac{1}{2t} \int_{B(x,t)}e^{ity}f(y)dy $$
(1) $u(x,t)$ is continuous on ...
0
votes
2answers
45 views
$ \lim\limits_{p\to +0}\int_X |f|^p d\mu = \mu(\{ x\in X | f(x) \neq 0\} $ [duplicate]
Measure space $(X,\mathcal{M},\mu)$, $f$ is integrable on $X$. Prove
$ \lim\limits_{p\to +0}\int_X |f|^p d\mu = \mu(\{ x\in X | f(x) \neq 0\} $
0
votes
1answer
25 views
Lebesgue measure space
Measure space $(X,\mathcal{B},\mu)$, $B_n\in \mathcal{B}$, $\mu(B_n) \le \infty, n=1,2,...,$.
(1) If $\sum\limits_{n=1}^{\infty} \mu(B_n) \le \infty $, then $\mu(\lim\sup_{n \to \infty} B_n)=0$.
(2) ...
2
votes
2answers
97 views
Solve limits in Lebesgue integral
Solve the limits of below:
(1) $\lim\limits_{n \to \infty} \int_0^n (1+\frac{x}{n})^n e^{-2x}dx$.
(2) $\lim\limits_{n \to \infty} \int_0^n (1-\frac{x}{n})^n e^{\frac{x}{2}}dx$.
(3) ...
-2
votes
2answers
49 views
Lebesgue conduct integral
I. Suppose $f\in \mathcal{L^1}(R^n),g\in \mathcal{L^1}(R^n)$, then conduct integral $f*g$ is defined as $f*g(x)= \int_{R^n}f(x-y)g(y)dy$ for all $x$.
My task is to prove following statements.
(1) ...
0
votes
1answer
39 views
Lebesgue integral of equal a.e. funtions
Suppose that $f : \mathbb{R} \to \overline{\mathbb{R}}$ is a Lebesgue integrable function. Assume that $g = f$ a. e. Show that $g$ is also Lebesgue integrable and that $\int f = \int g$.
Could I have ...
1
vote
1answer
42 views
Inf and sup for Lebesgue integrable functions
Let $D \subset \mathbb{R}$ be a measurable set of finite measure. Suppose that $f : D \to \mathbb{R}$ is a bounded function. Prove that
$$\sup\left\{\int_D \varphi \mid \varphi \leq f \text{ and } ...
1
vote
1answer
32 views
Lebesgue integral convergence
Suppose that $f : [0, 1] \to \mathbb{R}$ is Lebesgue integrable. Show that $x^n f(x)$ is also Lebesgue integrable on $[0, 1]$ for every $n \geq 1$ and compute $\lim_{n \to \infty} \int_{[0, 1]} x^n ...
1
vote
1answer
38 views
$f$ integrable $\implies g(x) = \int_{-\infty}^x f$ is absolutely continuous
Suppose that $f : \mathbb{R} \to \overline{\mathbb{R}}$ is an integrable function. Show that the function $g : \mathbb{R} \to \mathbb{R}$ defined by $g(x) = \int_{-\infty}^x f$ is absolutely ...
1
vote
1answer
39 views
Bounded integrable function
Let $f : \mathbb{R} \to \overline{\mathbb{R}}$ be an integrable funtion. Given $\varepsilon > 0$ show that there is a bounded integrable function $g$ such that $\int |f - g| < \varepsilon$.
I ...
3
votes
1answer
48 views
Measurability of a function defined on a product measure space, and related to a measurable function
Let $ (X,\mu) $ be a standard measure space - so that we may assume that $X$ is the unit interval $[0,1]$ with the Borel $\sigma$-algebra. Consider $X \times X$ with the product measure $\mu \times ...
5
votes
1answer
50 views
$L^{p}$ functions from Rudin Exercises 3.5
I am attempting a question from Rudin's "Real and Complex Analysis" Chapter 3 question 5. I shall summarise the question as below: Suppose that $f$ is a complex measurable function on $X$, $\mu$ a ...
1
vote
1answer
34 views
I did Lebesgue integration and $1/2$ appeared unexpectedly.
I am interested in Lebesgue integral over $[0;1]$ of the function
$$f(x) = \sum_{n=1}^\infty n \cdot \chi_{[0;n^{-2}]}(x);$$
Here $\chi_{[0;n^{-2}]}(x)$ is $1$ if $x \in [0;n^{-2}]$ and $0$ ...
1
vote
2answers
63 views
Lebesgue integral example
please help me with the Lebesgue integration on the function
$$
f(x)=
\begin{cases}
0 & \text{if $x$ < 0}\\
1/2 & \text{if $x$ = 0}\\
1 & \text{if $x$ > 0}
\end{cases}
$$
$$ ...
2
votes
1answer
51 views
Measurable functions on product measures
Let $ (X,\mu) $ be a measure space, and consider $X \times X$ with the product measure $\mu \times \mu $. Consider two functions $f$ and $g$ defined on $X \times X$ such that:
$f$ is measurable.
For ...
2
votes
1answer
56 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
Lebesgue Integral defined on infinite measure
Royden's Real Analysis Question: Let {$a_n$} be a sequence of nonnegative real numbers. Define the function $f$ on $E=[1,\infty)$ by setting $f(x)=a_n$ if $n\leq x< n+1$. I want to show that ...
2
votes
1answer
63 views
Verifying Fatou's Lemma
Royden's Real Analysis Question: Let {$f_n$} be a sequence of nonnegative measurable functions on $R$ such that $f_n\implies f$ pointwise on $E$. Let $M\geq0$ be such that $\int_Ef_n\leq M$ for all ...
0
votes
1answer
44 views
Absolute Convergence of a Function
I have got stuck with a question. Please help me.
Prove that $\dfrac{\sin(x)}{x}$ belongs to $L^p$ for all $p>1$.
Thank You.
1
vote
0answers
78 views
Are continuous functions strongly measurable?
Measure theory is still quite new to me, and I'm a bit confused about the following.
Suppose we have a continuous function $f: I \rightarrow X$, where $I \subset \mathbb{R}$ is a closed interval and ...
1
vote
0answers
44 views
Product of integrable functions also integrable? [duplicate]
I can't find in any book on measure and integral whether a product of two Lebesgue integrable functions is also integrable. Could someone clarify for me under what circumstances it is true?
To be ...
2
votes
2answers
60 views
the phrase “its derivative” when a function has a derivative only almost everywhere
Theorem: A function $F$ is an indefinite integral if and only if it is absolutely continuous
Corollary: Every absolutely continuous function is the indefinite integral of its derivative.
(thm ...
0
votes
1answer
65 views
Typical problem in functional analysis
I need help with this problem from my homework
Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ and let $\kappa : \Omega\rightarrow\mathbb{R}$ be a continuous function, such that there's ...
0
votes
1answer
40 views
Similar textbook to Konigsberger's Analysis 2?
I am currently taking an introductory course to real analysis and my professor has decided to leave Rudin's "Principles of Mathematical Analysis" when teaching us the concepts of Lebesgue integration. ...
2
votes
2answers
43 views
How to compute the limit of the following integral?
Given $b > 0$, let $g(x)$ be a continuous function defined on $[-b, b$]. What is the following limit?
$\lim_{N \rightarrow \infty} \frac{1}{\sqrt{N}} \int_{-b}^b e^{-\frac{Nx^{2}}{2}}g(x)\,dx $
...
2
votes
1answer
37 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 ...
0
votes
0answers
56 views
Change of differentation and integration signs.
I'm going through an old exam in a course I'm taking. I have the given rule:
Let $X$ be a measure space, $U$ be open subset in $\mathbf{C}$ and $f: U \times X \to \mathbf{C} $ be a function s.t. the ...
