# Tagged Questions

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### $\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$ for $f\in L^{p}$, $p \in [1,\infty)$

For $f\in L^{p}$, $p \in [1,\infty)$ we want to prove: $$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0$$ I'm not sure whether we can exchange the limit and the integral, cuz I cannot find ...
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### Application of Dominated Convergence Theorem.

Find $L=\lim\limits_{n \to \infty} \int_0^{n a} \exp\left(-\dfrac{t}{1+\frac{b t}{n}}\right) dt$, where $a>0$, $b>0$. I can't see what is dominating function, but I feel that I have to use ...
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### Limit Inf/Sup of Sequence of Set Example

In "A Probability Path", they have an example that states that the lim inf and lim sup of [0,n/(n+1)) is equal to [0,1). I guess I don't see how [0,1) is in all the sets except a finite number of ties ...
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### Proof: Tricky limit going to 0

I'm working on a proof and to complete it I need to find a way to choose an $n$ such that $(1-a)^n < \epsilon$ for a fixed $a$ such that $\frac12 < a < 1$ and any small $\epsilon$. I'm ...
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### Uniqueness of Weak Limit

As we know that weak limit of a sequence of Borel probability measures on metric space is unique. Do we have this property for general sequence of signed Borel measures on metric space? Thank you.
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### The limit of integral

Let $1 \le p < \infty$ and assume $f \in L^p(\mathbb{R})$. I'm trying to prove the limit of integral $$\lim_{x \to \infty} \int^{x+1}_x f(t)dt =0.$$ Can I use Riesz Theorem for Banach spaces?
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### Intuition behind $\limsup$ and $\liminf$ for probabilities

I've come across these limits in Fatou's lemma, this got me massively confused. I'd be grateful if someone could explain the intuition behind limit suprema and limit infima of probabilities (or ...
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### Lebesgue integral question concerning orders of limit and integration

I've got a hand-in question in a pure analysis course that I was hoping I might get a hint on - having difficulty coming up with a decent approach. The question: Let $(X,\Sigma,\mu)$ be a measure ...
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### Lebesgue integral calculation help

I have this limit to evaluate $$\lim_{n \rightarrow +\infty} \int_{0}^{2} \arctan \left(\frac{1}{1+x^n}\right) dx.$$ I have no idea how to solve this homework problem. Help!
A sequence of sets is defined as $A_n=\{x \in [0,1] : |\sum_{i=0}^{n-1} 1_{[\frac{i}{2n},\frac{2i+1}{4n})} - 1_{[\frac{2i+1}{4n},\frac{i+1}{2n})}| \geq p\}$ for some positive $p\geq0$. What is ...
I am looking for an intuitive explanation of $\liminf$ and $\limsup$ for sequence of sets and how it corresponds to $\liminf$ and $\limsup$ for sets of real numbers. I researched online but cannot ...