What measure should be used for this formula?

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.

N is 3

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.

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.

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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?