I got dual boot to windows and pop os, whenever i boot, it directly goes to pop os which is my priority os but it never asks me which is to boot up?

currently I have a question about the following point-set topology problem. Everything takes place in $(mathbb C,|cdot|)$ and we are given the following setup.

1. One has a sequence of closed connected sets of points in the complex plane ${L_t}_{t geq 0}$ (the precise definition for this problem should be irrelavant) such that $L_t subseteq L_{t+s}$ for each $s > 0$. So basically as $t$ increases the set grows aswell.
2. There exist continuous curves $gamma$ such that for each $t geq 0$ one has $gamma : [-t,t] to mathbb C$ such that $gamma([-t,t]) subseteq L_t$. Moreover we know especially that $gamma(t+s)$ and $gamma(-(t+s))$ are not in $L_t$ and together with another given property one can actually proof that $gamma(t)$ and $gamma(-t)$ are cut points of $L_{t+s}$ for $s > 0$. Using the definition found here this tells us (for example) that $L_{t+s} setminus gamma(t)$ is disconnected.

Question: We already have for each $t geq 0 : gamma([-t,t]) subseteq L_t$ I want to proof that one actually has equality, i.e. $L_t = gamma([-t,t])$.

Idea: Take for example $[a,b]$ in $mathbb R$ then every point $c in (a,b)$ is a cutpoint, the important take away here is that the ”end-points” of the interval $[a,b]$ are not cutpoints. Here something similiar should be possible. Since $gamma(t)$ is a cut point of $L_{t+s}$ it cannot be that that $L_{t+s} setminus L_t$ did not grow ”along” $gamma(t)$. Growing say along the middle of the curve around $gamma(t/2)$ should contradict some combination of the notion of cut point and continuity of $gamma$ but I find it hard to write down precisely.

Thanks in advance!

Let $(X,mathcal{E})$ and $(Y,mathcal{F})$ denote two measurable spaces and let $mu,nu$ denote two finite measures on $(X times Y, mathcal{E} otimes mathcal{F})$, where $mathcal{E} otimes mathcal{F}:= sigma(mathcal{E} times mathcal{F})$. Consider the claim

$$
mu(E times F) leq nu(E times F) text{ for all } (E,F) in mathcal{E} times mathcal{F} implies mu(A) leq nu(A) text{ for all } A in mathcal{E} otimes mathcal{F}.
$$

Question: Is this claim true?

Attempt:

Define the set $$
M = {A in mathcal{E} otimes mathcal{F} colon mu(A) leq nu(A)}.
$$
Then we want to show that $M = mathcal{E} otimes mathcal{F}$. Note that clearly $M subseteq mathcal{E} otimes mathcal{F}$. For the other inclusion, we have by assumption that $mathcal{E} times mathcal{F} subseteq M$ so it follows that $mathcal{E} otimes mathcal{F}= sigma(mathcal{E} times mathcal{F}) subseteq sigma (M)$ and hence it would suffice to show $M$ is a $sigma$-algebra.

However I am not sure this is the case. I can’t seem to show that $M$ is closed under complements and countable unions. In particular say for the union part, we can do the following:
$$
mu(bigcup_{n in mathbb{N}}A_n) leq sum_{n in mathbb{N}} mu(A_n) leq sum_{n in mathbb{N}} nu(A_n),
$$
but then we cannot go back to $nu(bigcup_{n in mathbb{N}}A_n)$. So this is where I am stuck.

I have also tried to look at Dynkin’s $pi$-theorem but at least I could not see how this was useful here.

Also I haven’t been able to come up with any counterexample yet.

As a final remark, I will just need to apply this result for $mathcal{E},mathcal{F} = mathcal{B(mathbb{R}^d}),mathcal{B(mathbb{R}^m})$ but I would like to prove it higher generality if this is indeed possible. If this is not possible, are there any conditions we can impose for it to hold? Any feedback/help is much appreciated!

In this paper1 in section 2, a method to write the elements of the coset of G/H is provided for GL(4), but I am interested in $GL^+$(2).

My matrix representation of $mathfrak{gl}(2)$ is

$$
begin{bmatrix}
a+x& -b+y\
b+y & a-x
end{bmatrix}
$$

My matrix representation of $mathfrak{so}(2)$ is

$$
begin{bmatrix}
0& -b\
b & 0
end{bmatrix}
$$

Reading the paper, it states that the element g of G can be decomposed as g=$gamma$h, where

$$
gamma = exp left( begin{bmatrix}
a+x& +y\
y & a-x
end{bmatrix} right) in G/H
$$

and where

$$
h = exp left( begin{bmatrix}
0& -b\
b & 0
end{bmatrix} right) in H
$$

Is this correct, or no?

I am suspicious of the argument, because to me

$$
exp left( begin{bmatrix}
a+x& y\
y & a-x
end{bmatrix} right)exp left( begin{bmatrix}
0& -b\
b & 0
end{bmatrix} right) neq exp left( begin{bmatrix}
a+x& -b+y\
b+y & a-x
end{bmatrix} right)
$$

Thus, $gamma h$ does not appear to realize all elements of $G$. Or do we not care about some missing elements for cosets?

This exercise specifically requires that we use the root test to determine whether the series converges or not.

All I’ve done so far is get the sequence in this form:
$$sqrt[n] frac{n^{n+frac{1}{n}}}{(n+frac{1}{n})^{n}} = sqrt frac{n^{(1+n^{-2})n}}{(n+frac{1}{n})^n}=frac{n^{1+n^{-2}}}{n+frac{1}{n}} = frac{n^{frac{n^2+1}{n^{2}}}}{n^{2}+1}$$

But, I’m not even sure if I’m on the right track here. Any guidance is appreciated.

Edit: I showed my effort. I didn’t tell anyone to solve it for me. Not sure why the downvotes..