# Tagged Questions

20 views

63 views

### Find a measurable set A$\subset(-1,1)$ such that $f(x)=m(A\cap(-1,x))$ has $f'(0)=1/2$

Now, I have done some work on this and by using the definition of the derivative, I come to the conclusion that if such a set $A$ exists, then I should have \begin{align*} ...
84 views

### Upper semicontinuity of a probability measure

Let $m$ be an atomless probability measure on $\mathbb{R}^m$. Consider $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ such that for all $v \in \mathbb{R}^m$, $x \mapsto f(x,v)$ is ...
111 views

### Limit of sequence of integrals

$$\lim_{n\rightarrow \infty} \int _0^\infty \frac{(1-e^{-x})^n}{1+x^2}dx.$$ this is less then $$\lim_{n\rightarrow \infty} \int _0^\infty \frac{1}{1+x^2}dx.$$ so the limit should exist by DCT. But ...
72 views

### Limit of a sequence of integrals

Determine $$\lim _{n \rightarrow \infty} \int_{-\infty}^\infty \frac{dx}{n(e^{x^2}-1) +1/n}$$ Since $e^{x} \geq t+1$ we know that $e^{x^2}-1 \geq 2t +t^2 \geq t^2$ if $t\geq 0$ So due to symmetry, we ...
147 views

### Why couldn't use the mathematical induction?

We can use mathematical induction which is deduced from Peano axioms and illustrated on Terence Tao's Real Analysis(here it is) Axiom 2.1 $0$ is a natural number. Axiom 2.2 If $n$ is a ...
81 views

### Continuity of $L^1$ functions with respect to translation

Let $f\in L^1$, consider the map $t\mapsto f_t=f(x-t)$, then how can one show that $t\mapsto f_t$ is continuous? More explicitly one wants to show that $\lim_{h\to 0}|f_{t+h}-f_t|_{L^1}=0$. I tried to ...
54 views

### Showing that the mean of translations of a function approaches 0 in $L_p$

Given $p \in (1,\infty)$, $f \in L^p(\Bbb R)$ and $T: \Bbb R \to \Bbb R,x \mapsto x+1$. How do I show that for $n \to \infty$ $$\frac{1}{n}\sum_{k=0}^n f \circ T^k \to 0$$ in $L^p$? I see that for ...
32 views

36 views

44 views

### How to understand stationary solution?

How to understand the stationary solution of the stochatic equation: $$X_{n+1}=A_n X_n+B_n$$ And where can I find more information?
84 views

### Limit problem for $L^p$ function

I am having problems with proving the following: Let $f$ be a $L^p$ function on $[0,1]$, $f:[0,1] \to \overline{\mathbb{R}}$. Prove that $$\lim_{t \to \infty} t^p \mu(x: |f(x)| \geq t) = 0.$$ ...
240 views

### Compute $\lim_{n\to\infty}\int_0^n \left(1+\frac{x}{2n}\right)^ne^{-x}\,dx$.

I'm trying to teach myself some analysis (I'm currently studying algebra), and I'm a bit stuck on this question. It's strange because of the $n$ appearing as a limit of integration; I want to apply ...
85 views

276 views

Consider $z \in \mathbb{R}^n$ and $\{ z_i \}_{i=1}^{\infty}$ with $z_i \rightarrow z$. Let $\phi: \mathbb{R}^n \times X \rightarrow \mathbb{R}_{\geq 0}$. $X$ is unbounded. I'm wondering if $$... 0answers 99 views ### \limsup bounded almost everywhere Consider z \in \mathbb{R}^n and a sequence \{ z_i \}_{i=1}^{\infty} such that z_i \rightarrow z. Let \phi: \mathbb{R}^n \times X \rightarrow \mathbb{R}_{\geq 0}. X is unbounded. I wonder ... 1answer 103 views ### Exercise: Limits and Probability Measure Let \mu be a probability measure on X (closed but unbounded), so that \int_X \mu(dx) = 1. Let functions f_i:X \rightarrow \mathbb{R}_{\geq 0}, i = 1,2,..., be Uniformly Integrable. Prove ... 1answer 78 views ### Sequences in a Probability Measure… Let \mu be a probability measure over the (closed but unbounded) set X \subseteq \mathbb{R}^m: \int_X \mu(dx) = 1. Consider function f:\mathbb{R}^n \times \mathbb{R}^n \times X \rightarrow ... 1answer 623 views ### Intuition behind \limsup and \liminf for probabilities I've come across these limits in Fatou's lemma, this got me massively confused. I'd be grateful if someone could explain the intuition behind limit suprema and limit infima of probabilities (or ... 1answer 270 views ### Lebesgue integral question concerning orders of limit and integration I've got a hand-in question in a pure analysis course that I was hoping I might get a hint on - having difficulty coming up with a decent approach. The question: Let (X,\Sigma,\mu) be a measure ... 1answer 301 views ### Lebesgue integral calculation help I have this limit to evaluate$$\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 ...