Explore Measurement Concepts with Our Examples and Questions

here is a question about Compressed sensing. Let us denote the $k$-sparse signal $x\in <!--\; \in \; -->{\mathbb{R}}^n$, measurement/sensing matrix $A\in <!--\; \in \; -->{\mathbb{R}}^{m\times <!--\; \times \; -->n}$ and the measurement $y=Ax\in <!--\; \in \; -->{\mathbb{R}}^m$.
In the noiseless situation, we know that the signal $x$ could be perfectly reconstructed if A satisfies the restricted isometry property. Denote the reconstructed signal as $x^{\prime }$. Now I am more interested in the reconstructed performance (i.e. mean square error between $x$ and $x^{\prime }$), instead of uniqueness or perfect reconstruction. It is intuitive that with more number of measurements $m$ the reconstruction performance would become better. But I could not find references for this idea. Is there any mathematical proof for this relationship?
Could anyone help me? Thanks in advance!

I'm trying to solve a problem that is similar to the fundamental theorem of calculus of variations.
Suppose $\mathrm{\varphi <!--\; \varphi \; -->}$ is a positive, continuous function on $[a,b]$ such that ${\mathrm{\varphi <!--\; \varphi \; -->}}^{1/n}\to <!--\; \to \; -->1$ pointwise. Suppose also that $f$ and $g$ are integrable on $[a,b]$ such that for all $n$,
${\int <!--\; \int \; -->}_{[a,b]}(f-<!--\; -\; -->g){\mathrm{\varphi <!--\; \varphi \; -->}}^{1/n}=0.(\ast <!--\; \ast \; -->)$
Show that $f=g$ almost everywhere.
My idea is to show that ${\int <!--\; \int \; -->}_{[a,b]}(f-<!--\; -\; -->g{)}^2=0$, from which the result follows. So I have, by Fatou's lemma:
${\int <!--\; \int \; -->}_{[a,b]}(f-<!--\; -\; -->g{)}^2={\int <!--\; \int \; -->}_{[a,b]}(f-<!--\; -\; -->g{)}^2\underset{n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}{\mathrm{\varphi <!--\; \varphi \; -->}}^{1/n}\le <!--\; \le \; -->\underset{n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim inf}{\int <!--\; \int \; -->}_{[a,b]}(f-<!--\; -\; -->g{)}^2{\mathrm{\varphi <!--\; \varphi \; -->}}^{1/n}$
And then I'm stuck at this point since I'm not sure how to evaluate this liminf. I want to say it is equal to 0, but I can't justify it.
Also, I haven't used above the fact (*). Any hints are appreciated. Thanks.

I have the following velocity measurements, where the sign of $V_e$ defines opposing directions of movement in a completely symmetric experimental setting:
$\begin{array}{|llr|}\hline V_e[cm/s]& V_m[cm/s]& \mathrm{\Delta <!--\; \Delta \; -->}V[cm/s]\\ 9& 9.38& 0.38\\ 8& 8.491& 0.491\\ 7& 7.482& 0.482\\ 6& 6.502& 0.502\\ 5& 5.726& 0.726\\ 4& 4.499& 0.499\\ 3& 2.021& -<!--\; -\; -->0.979\\ 2& 2.34& 0.34\\ 1& 2.018& 1.018\\ 0& 0& 0\\ -<!--\; -\; -->1& -<!--\; -\; -->0.501& -<!--\; -\; -->0.499\\ -<!--\; -\; -->2& -<!--\; -\; -->2.328& 0.328\\ -<!--\; -\; -->3& -<!--\; -\; -->2.988& -<!--\; -\; -->0.012\\ -<!--\; -\; -->4& -<!--\; -\; -->3.503& -<!--\; -\; -->0.497\\ -<!--\; -\; -->5& -<!--\; -\; -->4.506& -<!--\; -\; -->0.494\\ -<!--\; -\; -->6& -<!--\; -\; -->5.762& -<!--\; -\; -->0.238\\ -<!--\; -\; -->7& -<!--\; -\; -->7.479& 0.479\\ -<!--\; -\; -->8& -<!--\; -\; -->7.981& -<!--\; -\; -->0.019\\ -<!--\; -\; -->9& -<!--\; -\; -->8.496& -<!--\; -\; -->0.504\\ \hline\end{array}$
In this table, $\mathrm{\Delta <!--\; \Delta \; -->}V=\left|V_m\right|-<!--\; -\; -->\left|V_e\right|$
Running a Student t-test on $\mathrm{\Delta <!--\; \Delta \; -->}V$, we find that the mean does not significantly differ from zero, under a type I error of 5%. From this result, I conclude that $\mathrm{\Delta <!--\; \Delta \; -->}V$ is a random error.
The reviewer of my work (I'm an academic student) insists that my method does not account for the direction (i.e. the sign of $V$). That is indeed the case, since my test answers a precise question: Does the measurement method ($V_m$) systematically over/underestimates the true velocity ($||V_e||$)?
Instead, the reviewer uses $\mathrm{\Delta <!--\; \Delta \; -->}V=V_m-<!--\; -\; -->V_e$ to show that the method significantly overestimates $Ve$, especially when $V_e<0$, using the same t-test. However, I am having trouble finding what specific question such a test answers, and the reviewer's statement is wrong in my opinion.
What is the correct way of defining $\mathrm{\Delta <!--\; \Delta \; -->}V$ and discern a systematic measurement error?

I have seen this line in a computation and I wonder if that makes sense : $\mathbb{E}(X|Y)=\int <!--\; \int \; -->\mathbb{E}(X|Y=y)dP(Y=y)$ where P is a probability measure, $X,Y$ are random variables, so is $\mathbb{E}(X|Y)$. But I'm not familiar with this notation. If this is right, how can I understand it ?

I was reading about the different metric vs. abstract open set definitions of topologies and wondered, whether it is possible to define a measure on (${\mathbb{R}}^n,{\mathcal{B}}^n)$, such that this measure is also a metric.
Of course, in $\mathbb{R}$ the Lebesgue measure $\mathrm{\lambda <!--\; \lambda \; -->}$ and the Euclidean distance d coincide, in the sense that for any interval [a,b), we have $\mathrm{\lambda <!--\; \lambda \; -->}[a,b)=d(a,b)$. But with $n\ge <!--\; \ge \; -->2$ this is obviously not the case.
I guess we would define a measure and then prove that it also satisfies the definition of a metric, but I don't know where to start in constructing such a measure.

A measurement of the traffic generated by the packet source indicates that the average traffic is $\mathrm{\lambda <!--\; \lambda \; -->}$ [packets/s] and maximum traffic is $\mathrm{\sigma <!--\; \sigma \; -->}$ [packets/s]. The classic exponential distribution is not appropriate for modeling such a source of traffic, because the exponential distribution contains only one parameter (and two parameters were measured).
To model such a source of motion, you can use the shifted exponential distribution which is described by two parameters ($\mathrm{\gamma <!--\; \gamma \; -->},\mathrm{\delta <!--\; \delta \; -->}>0$):
$f(\mathrm{\tau <!--\; \tau \; -->})=\{\begin{array}{cc}0& \mathrm{\tau <!--\; \tau \; -->}<d\\ \mathrm{\gamma <!--\; \gamma \; -->}{e}^{-<!--\; -\; -->\mathrm{\gamma <!--\; \gamma \; -->}(\mathrm{\tau <!--\; \tau \; -->}-<!--\; -\; -->d)}& \mathrm{\tau <!--\; \tau \; -->}\ge <!--\; \ge \; -->d\end{array}$
Random variable $\mathrm{\tau <!--\; \tau \; -->}$ is the time interval between successive packets.
1. Draw a distribution graph (1). What is the relationship between the maximum traffic and $\mathrm{\delta <!--\; \delta \; -->}$, for the distribution (#)?
2. Designate a mean value of the distribution (#).
3. Based on measurements of traffic sources $(\mathrm{\lambda <!--\; \lambda \; -->},\mathrm{\sigma <!--\; \sigma \; -->})$ select firstly $\mathrm{\delta <!--\; \delta \; -->}$ parameter for distribution (#), then $\mathrm{\gamma <!--\; \gamma \; -->}$ parameter for distribution (#).

We can see conditional independence for $D$ and $Y$ given $X$ means $E[D\mid <!--\; \mid \; -->Y,X]=E[D\mid <!--\; \mid \; -->X]$ or $E[Y\mid <!--\; \mid \; -->D,X]=E[Y\mid <!--\; \mid \; -->X]$ in the causal analysis. However, by Wiki the conditional independence is defined as $E[YD\mid <!--\; \mid \; -->X]=E[Y\mid <!--\; \mid \; -->X]E[D\mid <!--\; \mid \; -->X]a.e.$
I cannot bridge these definitions. Is one of them stronger than the other one?
Thanks a lot.

Given a set $\mathrm{\Omega <!--\; \Omega \; -->}$ with a $\mathrm{\sigma <!--\; \sigma \; -->}$-field F defined on it. Let B be a subset of $\mathrm{\Omega <!--\; \Omega \; -->}$ and define
$B\cap <!--\; \cap \; -->\mathcal{F}=\{B\cap <!--\; \cap \; -->F:F\in <!--\; \in \; -->\mathcal{F}\}$. Call this ${\mathcal{F}}_B$.
The claim is that this collection of sets is a $\mathrm{\sigma <!--\; \sigma \; -->}$-field and hence (B, ${\mathcal{F}}_B$) is a measurable space.
I don't see why ${\mathcal{F}}_B$ is a $\mathrm{\sigma <!--\; \sigma \; -->}$-field just because it is a subset of one. Any insight appreciated

A continuous function $f$ between two topological spaces $X$ and $Y$ is generally defined as follows $f:X\to <!--\; \to \; -->Y$ is such that for every open subset $V$ of $Y$, the set $f^{-<!--\; -\; -->1}$($V$) is an open subset of $X$.
In this definition only I have a confusion/doubt: it is not mentioned anywhere that $f$ is bijective. So how can we take the inverse of $f$ and define the continuity of $f$ ? Moreover, there is homeomorphism in which it is necessary for $f$ to be bijective and hence speaking of $f^{-<!--\; -\; -->1}$, in this regard, makes sense.
Also, measurable functions in measure theory are defined in a similar fashion to that of continuous functions in topology. Here also I am confused about the same thing! Please give some insights. Thank you in advance.

How does inclusion of the measurement error in the model, as
$Y_i+{\mathrm{\Delta <!--\; \Delta \; -->}}_i=b{X}_i+{\mathrm{\epsilon <!--\; \epsilon \; -->}}_i$
affect the standard error of least square estimators $\stackrel{^<!--\; ^\; -->}{b}$ of coefficients $b$?
If I obtain a least squares fit from $Y_i=b{X}_i+{\mathrm{\epsilon <!--\; \epsilon \; -->}}_i$ and then later want to incorporate $\mathrm{\Delta <!--\; \Delta \; -->}$, is it possible to modify my standard error of $\stackrel{^<!--\; ^\; -->}{b}$ to obtain the right value?
(You can assume that $\mathrm{\Delta <!--\; \Delta \; -->}\sim <!--\; \sim \; -->\mathcal{N}(0,{\mathrm{\sigma <!--\; \sigma \; -->}}^2)$.)

I need to find the volume of the intersection set $A\cap <!--\; \cap \; -->B$ whereby $A=\{(x,y,z)\in <!--\; \in \; -->\mathbb{R}|x^2+y^2+z^2\le <!--\; \le \; -->1\}$ and $B=\{(x,y,z)\in <!--\; \in \; -->\mathbb{R}|x^2+y^2\le <!--\; \le \; -->1/2\}.$
It is clear that $A$ represents the unit ball centered at the origin and $B$ represents the cylinder with radius $\frac{1}{\sqrt{2}}.$. I should at some point use the Fubini theorem. I am puzzled by intersection set. Can somebody provide a solution proposal or a comment? Thanks.

It appears that these notations are equivalent when referring to the measure with which a function $f(x)$ is integrated with respect to. It seems to me that the expression $\int <!--\; \int \; -->fd{P}_X$ is very clear once some measure theory is learned. In what contexts are the other notations useful or necessary?

Let $F$ be a non-Archimedean local field, and ${\mathrm{\mu <!--\; \mu \; -->}}_F$ a Haar measure on $F$. The space $C_c^{\mathrm{\infty <!--\; \infty \; -->}}(F)$ of locally constant functions of compact support is spanned by characteristic functions of the sets $a+{\mathfrak{p}}^b$, for $a\in <!--\; \in \; -->F,b\in <!--\; \in \; -->\mathbb{Z}$ and $\mathfrak{p}$ the maximal prime ideal of ${\mathcal{O}}_F$.
I'm trying to prove the following: let ${\mathrm{\Phi <!--\; \Phi \; -->}}_0,{\mathrm{\Phi <!--\; \Phi \; -->}}_1$ be the characteristic functions of ${\mathcal{O}}_F,a+{\mathfrak{p}}^b$, respectively. If
${\int <!--\; \int \; -->}_F{\mathrm{\Phi <!--\; \Phi \; -->}}_0(x)d{\mathrm{\mu <!--\; \mu \; -->}}_F(x)=c_0>0,$
then
${\int <!--\; \int \; -->}_F{\mathrm{\Phi <!--\; \Phi \; -->}}_1(x)d\mathrm{\mu <!--\; \mu \; -->}(x)=c_0{q}^{-<!--\; -\; -->b}$
where $q$ is the cardinality of the finite field ${\mathcal{O}}_F/\mathfrak{p}$ and $||x||=q^{-<!--\; -\; -->v_F(x)}$ for all $x\in <!--\; \in \; -->F$. I feel to prove this I must express ${\mathcal{O}}_F$ in some way related to the sets $a+{\mathfrak{p}}^b$, and use properties of the Haar measure to find the measure of $a+{\mathfrak{p}}^b$. Can it be answered like this, or is there another way?

I really could use a hand on this question
Let $f:\mathbb{R}\mapsto <!--\; \mapsto \; -->f(\mathbb{R}):=E$, a continuous function. Let ${\mathrm{\mu <!--\; \mu \; -->}}_e$ be a probability on $(E,B(E))$.
Show that there exists a measure $\mathrm{\nu <!--\; \nu \; -->}$ on $\mathbb{R}$ such that :
$\mathrm{\forall <!--\; \forall \; -->}B\in <!--\; \in \; -->B(E),{\mathrm{\mu <!--\; \mu \; -->}}_e(B)=\mathrm{\nu <!--\; \nu \; -->}(f^{-<!--\; -\; -->1}(B))$
I really can't think of a single way to approache this. The obvious choice would seem to be $\mathrm{\nu <!--\; \nu \; -->}(B)={\mathrm{\mu <!--\; \mu \; -->}}_e(f(B))$, but this doesn't seem to work. Another idea would be to construct an outer measure with one of the methods available but I don't know how I would go about it
I thank you all in advance

Let $A$ then $A^c={\cup <!--\; \cup \; -->}_{n=i}^n{A}_i$, where $A_1,A_2,...,A_n$ are disjoint sets in $R$.
Since $A,A_i\in <!--\; \in \; -->R$ and $R$ is closed under finite intersection then $A\cap <!--\; \cap \; -->A_i\in <!--\; \in \; -->R$.
Hence $\mathrm{\varphi <!--\; \varphi \; -->}=A\cap <!--\; \cap \; -->A^c={\cup <!--\; \cup \; -->}_{n=i}^n(A\cap <!--\; \cap \; -->A_i)$, where $A\cap <!--\; \cap \; -->A_i$ are disjoint sets in $R$.
Since $\mathrm{\varphi <!--\; \varphi \; -->}$ is a finite union of disjoint sets in $R$, then ${\mathrm{\varphi <!--\; \varphi \; -->}}^c=\mathrm{\Omega <!--\; \Omega \; -->}\in <!--\; \in \; -->R$.
But now how to prove that $\mathrm{\varphi <!--\; \varphi \; -->}\in <!--\; \in \; -->R$?

Is there some inequality which relates
${\int <!--\; \int \; -->}_{\mathrm{\Omega <!--\; \Omega \; -->}}|f{|}^2\text{ and }{\int <!--\; \int \; -->}_{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\Omega <!--\; \Omega \; -->}}|f{|}^2$
for some $f\in <!--\; \in \; -->L^2(\mathrm{\Omega <!--\; \Omega \; -->})\cap <!--\; \cap \; -->L^2(\mathrm{\partial <!--\; \partial \; -->}\mathrm{\Omega <!--\; \Omega \; -->})$?

Given: Let $X$ be a sample from $P\in <!--\; \in \; -->\mathcal{P}$, ${\mathrm{\delta <!--\; \delta \; -->}}_0(X)$ be a decision rule (which may be randomized) in a problem with ${\mathbb{R}}^k$ as the action space, and $T$ be a sufficient statistic for $P\in <!--\; \in \; -->\mathcal{P}$. For any Borel $A\subset <!--\; \subset \; -->{\mathbb{R}}^k$, define
${\mathrm{\delta <!--\; \delta \; -->}}_1(T,A)=E[{\mathrm{\delta <!--\; \delta \; -->}}_0(X,A)|T]$
Let $L(P,a)$ be a loss function. Show that
${\displaystyle \int <!--\; \int \; -->L(P,a)d{\mathrm{\delta <!--\; \delta \; -->}}_1(X,a)=E[\int <!--\; \int \; -->L(P,a)d{\mathrm{\delta <!--\; \delta \; -->}}_0(X,a)|T]}$
My idea of proving this is to show that this holds for a simple function $L$ and generalize that to non-negative functions $L$ by using the conditional Montone Convergence Theorem. But I can't really show the equality for a simple function $L$. That is, if we take ${\displaystyle L=\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{c}_i{\mathbb{1}}_{A_i}}$ can I show that the given result indeed holds true?

Fix some probability space $(\mathrm{\Omega <!--\; \Omega \; -->},\mathcal{F},\mathbb{P})$, and let $\mathcal{A}\subset <!--\; \subset \; -->\mathcal{F}$ be a sub-$\mathrm{\sigma <!--\; \sigma \; -->}$-algebra. Suppose that $g,h:\mathrm{\Omega <!--\; \Omega \; -->}\to <!--\; \to \; -->\mathbb{R}$ are two $\mathbb{P}$-integrable and $\mathcal{A}$-measurable functions.
I need help understanding why $\{h>g\}\in <!--\; \in \; -->\mathcal{A}$? I understand that since $g$ and $h$ both are $\mathcal{A}$-measurable, it holds that for all $B\in <!--\; \in \; -->\mathcal{B}(\mathbb{R})$:
$h^{-<!--\; -\; -->1}(B)\in <!--\; \in \; -->\mathcal{A},g^{-<!--\; -\; -->1}(B)\in <!--\; \in \; -->\mathcal{A}.$
However, I can't seem to make the connection as to why $\{h>g\}\in <!--\; \in \; -->\mathcal{A}$ also holds true.

I have two probability density functions such that
$1={\int <!--\; \int \; -->}_a^b{\text{pdf}}_1(x)dx={\int <!--\; \int \; -->}_a^b{\text{pdf}}_2(x)dx$
which represent the same underlying process (i.e. obtained from different measurements).
I am more confident in the values of ${\text{pdf}}_1$ for values of $x<c$ where $a<c<b$ and more confident in values of ${\text{pdf}}_2$ where $c<x$. Is it possible to combine these pdfs to one while keeping this "confidence" scheme in mind? (Assuming that I have only access to the pdfs - not an assumption on the underlying process).
I can definitely "average/combine" the pdfs as a whole, but is it possible to be more confident in a section? The value of a range of the pdf is really meaningless without knowing the rest of the distribution, so I'm unsure if I can really proceed post-measurement.

Let $f\in <!--\; \in \; -->L^p(\mathbb{R})$ for some $1\le <!--\; \le \; -->p<\mathrm{\infty <!--\; \infty \; -->}$. Given $t\in <!--\; \in \; -->\mathbb{R}$, we set $f_t(x):=f(x-<!--\; -\; -->t)$.
Here I'm not sure if we can view $\{f_t\}$ as a sequence of functions, since they are not enumerated in a discrete sense. Is it true that when $t\to <!--\; \to \; -->0$, we have $f_t\to <!--\; \to \; -->f$ pointwise a.e. and $||f_t|{|}_p\to <!--\; \to \; -->||f|{|}_p$?
I'm also not sure if it's possible to apply results such as Dominated convergence theorem and Fatou's lemma when the functions are indexed like this, even if we were to set $g_t(x):=f(x-<!--\; -\; -->1/t)$ and let $t$ tend to infinity.