hosting – How can I measure website’s performance from various locations?

I’m looking for a web hosting service which has servers in various locations in order to host a multilingual website, but my question is as titled. I can measure a performance from my location (country) but want to do the same thing from different location. Is there any solutions? I’m planning to use Lighthouse for measurement but appreciate for another ideas.

measure theory – $int_Gg(x)f(x) ~ mathrm{d}mu(x)=0 ~ text{for all }gin L^1(G)$ implies $f=0$

My Definitions. Let $G$ be a locally compact Hausdorff group and $mu$ be a Haar measure on $G$. We have defined “Haar” measure as follows:

(Haar measure) It’s a nonzero left invariant outer Radon measure on $G$. An outer Radon measure is a locally finite Borel measure on $G$ which is outer regular on Borel sets and inner regular on open sets

(Assuming this definition) the author of the text (I think?) have used the following statement without proof. But I’m not very sure how exactly to prove this:

Statement. Let $f in L^1(G)$ such that
$$
int_Gg(x)f(x) ~ mathrm{d}mu(x)=0 ~ text{for all }gin L^1(G)
$$

Then $f=0$.

I tried to proceed with the same line of argument given here: since each compact set has finite measure, the assumption on $f$ implies that
$$
int_K f(x)~mathrm{d}mu(x)=0 ~text{for all compact }Ksubset G ~~~~~~~ (*)
$$

Case 1. $fin C_c(G)$ and $f$ is $Bbb{R}$-valued.

In this case I have managed to prove $f=0$.For this let $S:={fne 0}={f>0}cup{f<0}subset G$. With contrary if $mu(S)>0$ then WLOG $mu({f>0})=cup_n {f>frac{1}{n}}=:cup_n E_n>0$ hence there must exists $Nge 1$ s.t. $mu(E_N)>0$. Since $f$ is continuous so $E_N$ is open in $G$. Now I’ll use the (weak) inner regularity of $mu$ to arrive at a contradiction: if $K$ be a compact subset of $E_N$ with $mu(K)>0$ then we get
$$
0=int_K f(x) ~mathrm{d}mu(x)ge int_K frac{1}{N} ~mathrm{d}mu(x)=frac{1}{N}mu(K)>0
$$

a contradiction, hence $f=0$ in this case.

Case 2. $fin C_c(G)$

This follows from Case 2.

Case 3. $fin L^1(G)$.
Here I got stuck, I know $C_c(G)$ is dense in $L^1(G)$. So there would exists a sequence of members in $C_c(G)$ converging towards $f$. But the members of that sequence need not satisfy $(*)$. Also even such sequence exists I don’t know how to interchange the “limit” with “integral”??

Is the original statement valid only for continuous $f$?
Thank you.

measure theory – Question about a proposition in Munkres’s Analysis on Manifolds

I am reading through Munkres’s Analysis on Manifolds, and I get stuck in a proof of the lemma 18.1, that is stated as following:

Lema 18.1 Let $A$ be open in $mathbb{R}^n$; let $g:Ato mathbb{R}^n$ be a function of class $C^1$. If the subset $E$ of $A$ has measure zero in $mathbb{R}^n$, then $g(E)$ has measure zero in $mathbb{R}^n$.

He made out its proof in three steps. The first and second step are mentioned in the third, where he actually prove the theorem. Let me add some pictures of the third step.

enter image description here

enter image description here

(If you need pictures of the other two steps in order to solve the question above, let me know, please)

Note: A $delta$-neighborhood of a set $X$ is the union of all open cubes (in this case) with width $delta>0$ and centered at $xin X.$ The theorem 4.6 in that book states that every compact set $K$ that is contained in an open set $Usubset mathbb{R}^n$ has a $delta$-neighborhood contained in $U$.

So, the problem is here: When he covers the set $E_k$ by countably many cubes $D_i$ with certain properties, he asserts: Because $D_i$ has width less than $delta$, it is contained in $C_{k+1}$.

Why this is true? I mean, if each cube $D_i$ is centered at some point lying at $C_k$ it is clearly true, but we don’t know if this happens. I tried to give a proof that we can assume that each $D_i$ can be choosen in a way that is centered in $C_k$ but I couldn’t prove that.

Can you help me to justify that assertion on the book? Thanks in advance.

mg.metric geometry – Do Compact Universal Covers have Concentration of Measure phenomenon?

I have a sequence of compact Riemannian manifolds $M_n$ with $diam (M_n) to 0$ and finite fundamental groups $pi_1 (M_n)$ so that their universal covers $L_n$ are compact. Suppose $diam ( L_n ) = 1$ for all $n$, and their volumes are normalized so that $ vol (L_n)=1$.

Is it true that the family of metric measure spaces $L_n$ satisfies some concentration of measure phenomenon? Are they a Levy family? This is closely related to this other question: Diameter of universal cover .

pr.probability – Conditional Independence in Measure theoretic terms

Let $Omega$ be a compact Hausdorff space in $mathbb{C}^n$. Let $sigma_Omega$ be the Borel sigma algebra on $Omega$. Let $zeta: Omegalongrightarrowpartial mathbb{D}$ be a non constant continuous function. Let $sigma_{partial mathbb{D}}$ be the Borel sigma algebra on $partial mathbb{D}$(Unit circle on the complex plane). Now consider the sigma algebra $sigma_zeta={{zeta}^{-1}(A): ;Ain sigma_{partial mathbb{D}}}subset sigma_Omega$.

Now let $fin L^1(Omega, sigma_Omega, mu)$ and lets define a new measure $f_mu$ on $(Omega,sigma_zeta)$ as $f_{mu}(A)=int_A f dmu$. It is easy to see that for $Ain sigma_zeta $, ${mu}(A)=0$ implies $f_{mu}(A)=0$, i.e $f_{mu}(A)$ is absolutely continuous with the restriction of $mu$ to $sigma_zeta$, so by the Radon Nikodym theorem there exists a $gin L^1 (Omega, sigma_zeta, mu)$ such that
$int_A f dmu =int_A g dmu$ for every $Ain sigma_zeta$. Lets call this $g$ as the conditional expectation of $f$ and denote it as $E(f|sigma_zeta)$.

Can anyone explain me conditional independence of any two functions $h,kin L^1(Omega, sigma_Omega, mu)$ given $zeta$.

I need to understand this result in measure theoretic sense. or any reference for the same will be really appreciated.

measure theory – Show that $T_{alpha}$ is ergodic iff $alpha$ is irrational.

For any fixed $alpha in mathbb R$ define the function $T_{alpha} : (0,1) longrightarrow (0,1)$ by $x mapsto x + alpha (text {mod} 1),$ $x in (0,1).$ Show that $T_{alpha}$ is ergodic (i.e. if $T^{-1} (B) = B,$ for some Lebesgue measurable set $B$ then $mu(B) in {0,1},$ where $mu$ is the Lebesgue measure on $(0,1)$) iff $alpha$ is irrational.

I have tried to prove that but I failed. Could anybody kindly help me in this regard?

Thanks for your time.

Is there a tool to measure Google Ads investment by domain?

Hello,

I am looking for a tool that can tell me an estimate of the investment of adwords for a given domain. I tried spyfu.com and it kind of gives an approximate, but is there any other better than that? Is SEM Rush good? I would rather a free tool.

Thanks-

natural language processing – Cosine vs Manhattan to measure Text Similarity after doc2vec Embedding

I have a list of documents and made doc2vec embedding to them using BERT model so now they are stored as vectors in Elasticsearch database. I want to apply semantic search given a query from the user i should about the most n similar documents to this query. Elasticsearch gives similarity function options like Euclidean, Manhattan and cosine similarity . I have tried them and both Manhattan and cosine gives me very similar and good results and now i don’t know which one should i choose ? i made a lot of research and i found that both are good at high dimensions vector (Vectors are 768 dim)

bitcoin core – Had a question on using on-Chain Data(Glassnode) to measure Unrealised/Realised profit and loss of BTC holders – isn’t on-chain data not complete?

https://twitter.com/Negentropic_/status/1408436526262988801

The link above shows this tweet and pic: enter image description here

Isn’t on-chain data not the full universe of transactions?

for eg. Example 1: we have 10 users each holding one distinct BTC address. They all use Binance CEX (Centralized exchange) to buy/sell BTC. once they deposit their BTC into Binance CEX and start buying /selling BTC, the trail does dark as these Transactions are not on-chain meaning we don’t see their UTXO created prices and realised prices.

Example 2: Similar to Example 1 , but instead of depositing BTC, they deposit Fiat (USD) in Binance CEX to buy BTC. any buy/sell txns are on CEX so on chain data does not actually capture each user’s realised Profit and Loss

How do we then account for these user’s (as per example) Unrealized Profit and Loss if we don’t see their on-chain behavior since all the trading they do is on a CEX ??

pr.probability – Lemma 3.10 of paper ‘Periodic nonlinear Schrodinger Equation and Invariant measure’ by J.Bourgain

I am reading a paper ‘Periodic nonlinear Schrodinger Equation and Invariant measure’ by J.Bourgain. And I have two questions on the lemma 3.10.

enter image description here

  1. My first question is to get (3.12).

What I have up to now is as below.

begin{align*}
lambda sigma_M &< left| sum_{n~ M} frac{g_n(omega)}{M} e^{inx} right|_p \
&leq left| sum_{n~ M} frac{g_n(omega)}{n} e^{inx} right|_p\
&leq M^{frac{1}{2}-frac{1}{p}} left| sum_{n~ M} frac{g_n(omega)}{M} e^{inx} right|_2\
&leq M^{frac{1}{2}-frac{1}{p}} B
end{align*}

Thus, we get
$$sigma_M frac{lambda}{B}<M^{frac{1}{2}-frac{1}{p}}$$

In order to have (3.12), I would need to remove $sigma_M$ by using (3.14) but it seems like I went too far to use it. I will be happy to have any idea on this inequality.

  1. My one another question is if there is any reference to understand this line in the proof of same lemma.

enter image description here

Thanks in advance.