All Questions
28
questions
2
votes
0
answers
43
views
A question regarding bounded variation and differentialibility as well as integration
Let $a, b \in \mathbb{R}$ with $a < b$, and $f: [a, b] \to \mathbb{R}$ be a monotonically increasing, right-continuous function. Show that there exists a monotonically increasing function $g: [a, b]...
- 327
0
votes
0
answers
58
views
Question regarding derivative and measures
Here's what I need to show:
Let $\mu$ be any finite signed Borel measure on $[-\pi,\pi]$. Let $g: [-\pi,\pi] \to \mathbb R$ be defined by $g(\theta ) = \int_{0}^{\theta} d\mu (t)$ if $\theta >0$ ...
- 4,015
2
votes
0
answers
237
views
Measure theoretic proof of Leibniz Rule for Differentiation under the Integral with varying integration limits
I'm searching for a rigorous proof of Leibniz rule of integration for varying integration limits, i.e.:
$$
\frac{d}{dx}\int^{b(x)}_{a(x)} f(x,t) dt = f(x,b(x)) \frac{db(x)}{dx} -
f(x,a(x)) \frac{da(x)}...
- 3,309
0
votes
1
answer
722
views
Monotonic function that is not of bounded variation [closed]
Are all monotonic of bounded variation, or would there exist a counterexample of a function that is monotonic but not of bounded variation?
user917822
0
votes
1
answer
144
views
Is the Hardy-Littlewood Maximal Function of Increasing Nonnegative Functions Increasing As well?
If we assume that the function $f$ is increasing, nonnegative and integrable on $\mathbb{R}$ can we conclude that the Hardy-Littlewood maximal function of this function is increasing as well?
I am ...
1
vote
1
answer
112
views
The Fundamental Theorem of Lebesgue Calculus
Let $[a, b]$ be an interval of $\mathbf{R}$. If $\varphi:[a, b] \rightarrow \mathbf{R}$ is continuous, then the function $F:[a, b] \rightarrow \mathbf{R}$ defined by $F(x)=\int_{[a, x]} \varphi(x) d \...
- 1,589
1
vote
0
answers
152
views
Weak derivative under integral sign
Consider a locally integrable function $u$ on $\mathbb{R}^n$ (in $C^\infty$, for argument's sake). Then the function $f:[0,\infty)\times \mathbb{R}^n\rightarrow\mathbb{R}$ given by
\begin{equation}
f(...
- 1,425
6
votes
2
answers
297
views
Working with infinitesimals of the form d(f(x)), for example d(ax), and relating them to dx (integration, delta function)
I am trying to get a better understanding on how we can manipulate the infinitesimal dx in an integral $$\int f(x) dx$$
I have come across the following
$$ d(\cos (x)) = -\sin(x) dx$$
Therefore
$$\int^...
- 443
1
vote
0
answers
79
views
Bounded function that has antiderivative but is not integrable on $[a,b]$.
I am studying Riemann integration and I need to find a counterexample.I want a function that is bounded,and has an antiderivative on $[a,b]$ but is not Riemann integrable on $[a,b]$.I think it will ...
- 3,619
0
votes
1
answer
302
views
Differentiating Lebesgue Integral - Application in Economics
I am currently reading up on measure theory and Lebesgue integration in order to gain a somewhat more nuanced understanding of a few "aggregation-adjacent" results in economics.
The context of my ...
- 91
22
votes
3
answers
998
views
Derivative and calculus over sets such as the rational numbers
I am interested in the derivative of a function defined on a subset $S$ of $[0, 1]$. The subset in question is dense in $[0, 1]$ but has Lebesgue measure zero. My actual question can be found at the ...
- 3,645
0
votes
1
answer
45
views
$L^2$ boundness of derivatives of a uniformly convergent sequence?
Let $f_n\colon (a,b)\rightarrow \mathbb{R}$ be a sequence of functions in $C^\infty_c(a,b)$, that converges uniformily to a function $f$. Is the sequence of the derivatives bounded in $L^2(a,b)$? In ...
0
votes
1
answer
165
views
Why is it difficult to define integral although it's easy to define differentiation? [duplicate]
Defining multivariable differentiation is just linear algebra. However, defining integration is complicated measure theory. Why are these efforts so different?
- 659
0
votes
1
answer
223
views
Prove that $ \frac{d\nu}{d\lambda}=\frac{d\nu}{d\mu}\cdot \frac{d\mu}{d\lambda},\text{ } \lambda\text{-a.e.} $
Problem:
Let $ \nu $ be a signed measure and $ \mu,\lambda $ be measures on $ (\Omega,\mathcal{F}) $ such that $ \lambda,\mu,\nu $ are $ \sigma $-finite, $ \nu\ll\mu $ and $ \mu \ll \lambda $. Prove ...
- 849
8
votes
1
answer
2k
views
Differentiation under the integral sign for volumes in higher dimensions
Consider a smooth convex/compact domain $D\subset \mathbb{R}^n$ and a smooth, concave function $F:D\to \mathbb{R}$. Then we can define the function that simply takes the volume of the upper contour ...
- 3,380