## question

Is there a measure that can be associated with the notion of area (in the LHS), which is described by the formula:

$$lim_ {k to infty} lim_ {n to infty} sum_ {r = 1} ^ n a_r left (f ( frac {k} {n} r) frac {k } {n} right) = lim_ {s to 1} ! underbrace { frac {1} { zeta (s)} sum_ {r = 1} ^ infty frac {a_r} {r ^ s}} _ { text {removable singularity}} int_0 ^ infty f (x) , dx$$

Where $$f$$ is smooth, $$int_0 ^ infty f (x) , dx$$ is finite and $$lim_ {n to infty} frac { log ^ 2 (n)} {n} sum_ {r = 1} ^ n | b_r | = 0$$

Where $$b_r = sum_ {d midr} a_d mu ( frac {m} {d})$$, For the proof, see here

## The LHS

In the Riemann integral, the area is calculated essentially by dividing the area in $$N$$ rectangular stripes and then take $$N to infty$$ and $$a_r$$ is a real number.

Let's say I divide the area into several areas $$3$$ Strip, however, I tell the first strip $$a_1$$ times the $$2$$nd strips $$a_2$$ times and the third strip $$a_3$$ times.

The LHS describes the case $$N to infty$$ Then I go to the limit $$k to infty$$

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## Measure Borel sets and Method I

If $$(X, d)$$ is a metric space, say a function $$tau$$ to a class $$mathscr {C}$$ of subsets of $$X$$ is a premise if $$emptyset in mathscr {C}$$. $$tau ( emptyset) = 0$$ and $$0 le tau (C) le + infty$$ for all $$C in mathscr {C}.$$

If $$tau$$ is a premiere for a class $$mathscr {C}$$ of subsets of $$X$$, then the set function
$$mu (E) = inf left { sum_ {i = 1} ^ infty tau (U_i): U_i in mathscr {C}, E subset cup_ {i = 1} ^ infty U_i right }$$
is a measure on $$X.$$ The action is called Method that I measure, For more details, see C.A. Rogers & # 39; s book titled Hausdorff measure,

A literature claims that "Borel amounts are generally not measurable in terms of dimensions from Method I constructions". Can someone give an example? Thank you very much.

## powerbi – How do I calculate a measure to display the percentage for a clustered column chart, with the percentages for each cluster totaling 100%?

In Power BI, I want to create a clustered column chart that displays percentages where the percentages for each cluster sum to 100%.

The default setting for displaying percentages specifies percentages where the total for all clusters (not for each cluster) is 100%. I think I need to create a measure and use it in the diagram, but I can not work out the right formulas to do that.

Below is a code I tried. These do not work – they are displayed 100% for each column in the chart.

q13_Pct by Accommodation Type = DIVIDE (CALCULATE (COUNT (Accommodation_2019_Term1_Results (q13_labelled)), ALL (Accommodation_2019_Term1_Results (Accommodation__c__labelled))), CALCULATE (COUNT (Accommodation_2019_Term1_Results (Accommodation_2_labelled))

q13_Pct by Accommodation Type = DIVIDE (COUNT (Accommodation_2019_Term1_Results (instanceID)), CALCULATE (COUNT (Accommodation_2019_Term1_Results (instanceID)), ALL (Accommodation_2019_Term1_Results (q13_labelled)))

q13_Pct by Accommodation Type = DIVIDE (COUNT (Accommodation_2019_Term1_Results (instanceID)), CALCULATE (COUNT (Accommodation_2019_Term1_Results (instanceID)), ALL (Accommodation_2019_Term1_Results (Accommodation__c__labelled)))

I want to see a grouped column chart that displays percentages so that the columns in each cluster sum up to 100%.

The standard percentage option provided by Power BI returns percentages that are 100% for each column in the entire chart.

The above code (all) returns 100% for each column in the chart.

## Measure theory – dual of \$ L ^ 1 ( mu) \$ in general …

I wondered about the double of $$L ^ 1 ( mu)$$ when $$mu$$ is not $$sigma$$-at last.

I just got a copy of Cohn measurement theory, I see different results that show that the dual is $$L ^ infty$$ for some non-$$sigma$$-endliche dimensions, but that is simply not for a general measure; So far, I do not see anything about what the dual actually is is in general. I wonder:

Say $$g$$ is locally measurable if $$g chi_E$$ is measurable for every measurable quantity $$E$$ With $$mu (E) < infty$$ equivalent, if $$fg$$ is measurable for everyone $$f in L ^ 1$$).

Question. Accept $$mu$$ is a measure and $$Lambda in L ^ 1 ( mu) ^ *$$, Does it follow that there is a limited locally measurable function? $$g$$ With $$Lambda f = int fg$$?

It's probably a standard counterexample, and I'm just clueless again. But the few counterexamples to it $$(L ^ 1) ^ * = L ^ infty$$ that I know are no counter-examples to this:

Example. Say $$X$$ is an innumerable amount, $$A$$ is the algebra of countable or countable sets, and $$mu$$ counts measure $$X$$, limited to $$A$$, It's easy to see, though $$Lambda in (L ^ 1) ^ *$$ then there is a limited function $$g$$ With $$Lambda f = int fg$$, and here everyone Function is measurable locally. (If $$g$$ is then limited and not measurable $$f mapsto int fg$$ is an element of $$(L ^ 1) ^ *$$ that is not induced by one $$L ^ infty$$ Function.)

Warning: If, like me, you browse through Cohn without carefully reading the first few chapters, Proposition 3.3.5 will clearly be wrong:

Cohn 3.3.5, circumscribed. If $$mu$$ is a measure and $$g in L ^ infty ( mu)$$ then $$|| g || _ infty = || g || _ {(L ^ 1) ^ *}$$,

$$newcommand set (1) { {# 1 }}$$
Apparent counterexample. To let $$X = set {1,2}$$. $$mu ( set 0) = 1$$. $$mu ( set 1) = infty$$, To let $$g = chi _ { set 1}$$, Then $$|| g || _ infty = 1$$. $$|| g || _ {(L ^ 1) ^ *} = 0$$,

Resolution. Cohn $$|| g || _ infty$$ is not what you think. Say the measurable amount $$N$$ is local zero if $$mu (N cap E) = 0$$ for every measurable sentence $$E$$ With $$mu (E) < infty$$, He defines $$|| g || _ infty$$ to be the infimum of $$M$$ so that $$set {x: | f (x) | ge M}$$ is zero locally.

(If someone familiar with the book points to other similar sources of potential Cohnfusion, this would be very welcome …)

## Database Design – How to objectively measure the effectiveness of a data model?

I'm not sure which topic that is, but I'm researching the theory behind the following questions:

What is the best data model for a particular dataset (say, this dataset is large but set for simplicity)?

Are there some universal measures that optimize data compression?

Which variables affect the & # 39; effectiveness & # 39; a data model?

I have done some research on this subject, and often mention the concept of entropy in terms of information coding in general and text compression in particular. Was a similar topic investigated by someone in the world of database / content modeling?

All references would be very grateful.

## Invariance of density measure – Math stack exchange

If we go through spherical coordinates in $$mathbb {R} ^ n$$ it defines a density measure relative to the Lebesgue measure.

If I call $$mu$$ Is this metric invariant on rotation?

I know that the Lebesgue measure is not safe $$mu.$$

## Probability – existence and uniqueness of a stationary measure

The same question was also asked on MSE https://math.stackexchange.com/questions/3327007/existence-and- uniqueness-of-a-stationary-measure.

Recently I asked the following question about MO Attractors in Random Dynamics.

To let $$Delta$$ be the interval $$(-1.1)$$Then we can look at the probability space $$( Delta, mathcal {B} ( delta), nu)$$, from where $$mathcal {B} ( Delta)$$ is the Borel $$sigma$$algebra and $$nu$$ is equal to half the Lebesgue measure.

Then we can equip the room $$Delta ^ { mathbb {N}}: = {( omega_n) _ {n in mathbb {N}}; \ omega_n in Delta \ forall n in mathbb {N} }$$ with the $$sigma$$-Algebra $$mathcal {B} ( Delta ^ { mathbb {N}})$$ (Borel $$sigma$$-algebra of $$Delta ^ { mathbb {N}}$$ induced by the product topology) and the probability measurement $$nu ^ { mathbb {N}}$$ in measurable space$$( Delta ^ { mathbb {N}}, mathcal {B} ( Delta ^ { mathbb {N}}))$$, so that
$$nu ^ { mathbb {N}} left A_1 times A_2 times ldots times A_n times prod_ {i = n + 1} ^ { infty} Delta right) = nu (A_1) cdot ldots cdot nu (A_n).$$

Now let it go $$sigma> 2 / (3 sqrt {3})$$ be a real number and define
$$x _- ^ * ( sigma) = text {The unique real root of the polynomial} x ^ 3+ sigma = x,$$
$$x _ + ^ * ( sigma) = text {The unique real root of the polynomial} x ^ 3- sigma = x,$$
that's easy to see $$x _ + ^ * ( sigma) = -x _- ^ * ( sigma)$$,

We can then define the function
$$h: mathbb {N} times Delta ^ mathbb {N} times (x _- ^ * ( sigma), x _ + ^ * ( sigma)) to (x _- ^ * ( sigma), x _ + ^ * ( sigma)),$$
in the following recursive way,

• $$h (0, ( omega_n) _ {n}, x) = x$$. $$forall ( omega_n) _n in mathbb {N}$$ and $$forall x in mathbb {R}$$;
• $$h (i + 1, ( omega_n) _ {n}, x) = sqrt (3) {h (i, ( omega_n) _ {n}, x) + sigma omega_i}.$$

That's how we are for everyone $$x in mathbb {R}$$ and $$( omega_n) _n in Delta ^ mathbb {N}$$Define the following order
$$left {x, sqrt (3) {x + sigma omega_1}, sqrt (3) { sqrt (3) {x + sigma omega_1} + sigma w_2}, sqrt ( 3) { sqrt (3) { sqrt (3) {x + sigma omega_1} + sigma w_2} + sigma w_3}, ldots right }.$$

Now define the following family of Markov kernels
$$P_n (x, A) = nu ^ { mathbb {N}} left ( left {( omega) _ {n in mathbb {N}} in Delta ^ { mathbb { N}}; h (n, ( omega_n) _ {n in mathbb {N}}, x) in A right } right).$$

A probability measure $$mu$$ in the $$((x _- ^ * ( sigma), x _ + ^ * ( sigma)), mathcal {B} ((x _- ^ * ( sigma), x _ + ^ * ( sigma) ))$$ is a called stationary measure, if

$$mu (A) = int _ {(x _- ^ * ( sigma), x _ + ^ * ( sigma))} P_1 (x, A) text {d} mu (x) ; forall A in mathcal {B} ((x _- ^ * ( sigma), x _ + ^ * ( sigma))),$$
from where $$mathcal {B} ((x _- ^ * ( sigma), x _ + ^ * ( sigma)))$$ is the Borel $$sigma$$-Algebra. Besides, once $$(x _- ^ * ( sigma), x _ + ^ * ( sigma))$$ it is easy to prove that there is at least one stakionäre measure.

The answer I received to MO suggests that there is only one stationary measure.

Does anyone know if that's true? An indication of such a result is sufficient for my purposes.

## real analysis – Retract a finite measure

In a preliminary examination, I came across the following problem.

Let X and Y be compact metric spaces. Accept, $$phi: X to Y$$ is a continuous surjective map. To let $$D = {f in C (X): f (x) = f (x ') whenever phi (x) = phi (x') }$$

a) Show that D is a closed subspace of $$C (X)$$ and the $$D = {g circ phi: g in C (Y) }.$$

b) Leave $$nu$$ Be a finely positive Borel measure for Y. Prove that there is a finite positive Borel measure $$mu$$ on X so that $$mu ( phi ^ {- 1} (F)) = nu (F)$$ for all Borel sunsets $$F$$ from Y.

For part a) it is easy to prove that D is a closed subspace and also a function $$f$$ in D looks like $$g circ phi$$ for a few g. But I can not argue why g should be continuous.

My main problem concerns part b). While it's easy to show that $${ phi ^ {- 1} (F) }$$where F passes over Borel subsets of Y is a sigma algebra. And we can define a finite measure $$mu$$ to this sigma algebra of $$mu ( phi ^ {- 1} (F)) = nu (F)$$, I can show that this is well defined. My problem is that this sigma algebra is on $$X$$ can be much smaller than the Borel Sigma algebra on X. So do I need to extend this metric to the whole Borel Sigma algebra? If so, how can I do that? Also, I do not see any use of the surjectivity of $$phi$$ in part b). Am I right in my understanding?

## Presence of a positive measure square in a positive measure set in \$ mathbb {R} ^ 2 \$.

To let $$E subset mathbb {R} ^ 2$$ so that $$overline {E}$$ has a positive effect in $$mathbb {R} ^ 2$$, Can we always find $$A, B subset mathbb {R}$$ so that $$A × B subset E$$ and $$overline {A}, overline {B}$$ is of positive importance in $$mathbb {R}$$?