All Questions
45
questions
1
vote
1
answer
99
views
Show that $f(x):=e^{-x^{4}}$ is Lebesgue integrable
So I want to check if my solution is correct
I must show that the function $f: [-\infty,\infty] \rightarrow \mathbb{R}$ , $x\rightarrow e^{-x^{4}}$ is Lebesgue integrable.
For showing it I have ...
- 900
1
vote
1
answer
287
views
$f$ integrable iff set of discontinuities have measure $0$. But why is $\chi_\mathbb{Q}$ not integrable?
I have a theorem stating that if $f:[a,b] \rightarrow \mathbb{R}$ is bounded, then $f$ is Riemann integrable if and only if the set of discontinuities of $f$ has measure 0. Using this you can prove ...
- 391
0
votes
1
answer
485
views
Bounded function is Riemann-Integrable if and only if its upper and lower integrals are equal
I was looking at this proof from my measure theory class but specifically im having trouble understanding the $\Leftarrow$ part, that is, proving that for a bounded function $f:[a,b]\rightarrow \...
- 69
1
vote
1
answer
60
views
When is the indicator of a sublevel set integrable?
I have a smooth manifold (even embedded, if you wish) and a smooth real valued function $f:M\rightarrow \mathbb R$.
I am interested to know when are the indicators of sublevel sets of $f$ integrable (...
- 4,333
0
votes
1
answer
55
views
Calculating double integral via lebegue technique
I heard that one of the drawback of Riemann integral is;it is very laborious to find Riemann integral in higher dimensions e.g Calculating Riemann integral of bounded function $f:D\subset\mathbb{R^2}\...
- 733
1
vote
0
answers
154
views
Jordan measurability of "Area under the curve" of a function
Let $I = [a,b] \subset \mathbb{R}$. Let $f : I \to \mathbb{R}$ be a riemann integrable function on $I$. Consider the set $S_f$ given by, $$S_f = \{ (x,y) \in \mathbb{R}^2 | x \in I, 0 \leq y \leq f(x) ...
- 355
2
votes
1
answer
502
views
Expectation with Lesbesgue vs Riemann-Stieltjes integration
I have studied measure theory and probability theory before, and we defined the expectation of a random variable $X$ with respect to some probability space $(\Omega,\mathcal{F},\mathbb{P})$ as the ...
- 560
1
vote
1
answer
82
views
Integration over cross product and $\mu(x)$ as counting measure meaning in Lebesgue integral?
I was reading through this paper, and I encountered notation that is confusing to me. It's on reinforcement learning / Markov decision processes, but I aim to explain my question in a way that won't ...
3
votes
1
answer
166
views
Integration of a radial function over a bounded domain
Let $\Omega$ be a bounded domain in $\mathbb{R}^N$. Let $f(x)=|x|^\alpha$. Then $f\in L^1(\Omega)$ if $\alpha>-N$.
The above fact seems to hold for the following reason.
Since $\Omega$ is bounded, ...
- 431
1
vote
0
answers
184
views
On integral of a function that is zero except on a measure zero set.
I was/am trying to prove that if a function $f$ is Riemann integrable on $[a,b]$ and $f(x)=0$ on $[a,b]$ except on a subset $S$ of measure zero, then $\int_{a}^{b}{f(x)dx}=0$
I think I have come up ...
- 312
0
votes
0
answers
156
views
Is every odd function on a symmetric interval of $\mathbb{R}$ lebesgue integrable?
For example is an odd $f:\mathbb{R}\rightarrow \mathbb{R}$ over the entire real line Lebesgue Integrable? I know that its improper Riemann integral will always exist and equal zero so I am asking if ...
- 1,280
2
votes
1
answer
442
views
Definition of zero measure sets in a manifold
What show below is a reference from the text Analysis on Manifolds by James Munkres.
So with this results I like to discuss the following definition given by Munkres
So unfortunately I don't ...
- 4,906
2
votes
0
answers
163
views
If $S\subseteq\Bbb R^n$ is rectifiable then its projection $S'$ in $\Bbb R^{n-1}\times\{t\}$ is rectifiable too.
Definition
Let $A$ be a subset of $\Bbb R^n$. We say $A$ has measure zero in $\Bbb R^n$ if for every $\epsilon>0$ there is a covering $Q_1,Q_2,...$ of $A$ by countably many rectangles such that
$$
\...
- 4,906
1
vote
2
answers
61
views
Evaluate $\int_0^1f(t)dt$
Let $f(x)$ be an integrable function on $[0,1]$ that obeys the property $f(x)=x, x=\frac{n}{2^m}$ where $n$ is an odd positive integer and m is a positive integer. Calculate $\int_0^1f(t)dt$
My ...
- 849
1
vote
2
answers
310
views
The set $\Bbb R^{n-1}\times\{t\}$ has measure zero in $\Bbb R^{n}$ for any $t\in\Bbb R$
Definition
Let $A$ be a subset of $\Bbb R^n$. We say $A$ has measure zero in $\Bbb R^n$ if for every $\epsilon>0$ there is a covering $Q_1,Q_2,...$ of $A$ by countably many rectangles such that
$$
\...
- 4,906