Tagged Questions

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Derivation using Ito calculus?

I am reading the paper "Coupling Wiener processes by using copulas" by P. Jaworski and I've come across a statement I cannot reproduce. Let $L^{-}$ and $L^{+}$ be differential operators acting on ...
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Conditional probability explained?

Let $F_A$ be the CDF to the random variable $A$ ( and $B$ another independet rv), how do we get that $P(A+B \le s) = \int_{\mathbb{R}} P(A+B \le s\mid A=x ) \, dF_A(x)$ (This is probably a ...
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Are all singular functions of bounded variation?

Let $f$ be a function of bounded variation on $[a,b]$. Then there exist a unique pair (up to adding a constant) of absolute continuous function $g$ and singular function $h$ (i.e., $h'=0$ a.e.) such ...
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Does a nondecreasing, differentiable function have continuous derivative?

Are the following statements true? How to prove or disprove? (1). Let $f$ be a nondecreasing, differentiable function on $[0,1]$. Then $f$ is absolutely continuous? To be stronger, (2). Let $f$ ...
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Without Lebesgue

Everyone knows following problem. Let $f$ be positive function on $[0,1]$ and there exist $I = \int_{0}^{1}f(x)dx$. Prove that $I>0$. (recall that there are only two cases: $I=0$ or $I>0$. NOT ...
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$xf''(x) , xf', f \in L^{2}$ is $f' \in L^{1}$?

I am stuck on the following problem. I have a function $f$ such that $f$ is bounded on $(0,1)$, $xf'(x)$ is bounded on $(0,1)$, $f \in L^{2}(0,1)$, $xf' \in L^{2}(0,1)$, and $xf'' \in L^{2}(0,1)$. ...
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Topology of convergence in measure

Currently I am doing some measure theory (on $X=[0,1]$ with the Borel-Sigma algebra and the Lebesgue measure), and I am looking at sets $A \subset L^p$, such that for all $q \in (0,p)$, the topologies ...
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$f '$ is not Lebesgue integrable on $[-1,1]$

Let f be that function from R to R defined by f(x)= 0 if x=0 x^2 sin(1/x) if x not = 0 show that the function f' is ...
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Show that $\mu$ is unique and right-continuous

This problem is a little tricky, so I'd like some of your thoughts on it! Assume $\mu$ is a measure on $(\mathbb{R},\mathbb{B})$ where $\mathbb{B}$ are the Borel sets on $\mathbb{R}$. Also, ...
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$C^\omega(\Omega)\cap C^\infty_0(\Omega)$.

Let $\Omega$ denote an open connected set in $\mathbf{R}$ (AKA open interval). Is it true and how can we prove it that $C^\omega(\Omega)\cap C^\infty_0(\Omega)$ consists of the zero function alone, ...
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Why is $g_n:=\inf\{f_n,f_{n+1},…\}$ integrable if $f_n$ are?

This question is motivated by the proof of the Fatou's lemma. My text defines $g_n:=\inf\{f_n,f_{n+1},\ldots\}$ and states that it's Lebesgue integrable (each $f_n$ is). We proved that point-wise ...
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Does an absolutely integrable function tend to $0$ as its argument tends to infinity?

Suppose that $f:[0,\infty)\rightarrow\mathbb{R}$ is continuous. Is it true that $$\int_{0}^\infty|f(t)|dt<\infty\Rightarrow \lim_{t\rightarrow\infty}f(t)=0?$$ If so can you provide a proof, ...
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If a continuous function from $\mathbb{R}$ to $[0,\infty)$ does not tend to zero, is its integral greater or equal than some linear function?

Consider a continuous function $f:\mathbb{R}\rightarrow[0,\infty)$ that does not tend to zero as its argument tends to infinity. Formally, there is some $\varepsilon>0$ such that there does not ...