Fdtxjr

What are some good books on Machine Learning and AI like Krugman, Wells and Graddy's "Essentials of Economics"

Why was the shrinking from 8″ made only to 5.25″ and not smaller (4″ or less)?

Plagiarism or not?

Is it possible to create a QR code using text?

I would say: "You are another teacher", but she is a woman and I am a man

Apex Framework / library for consuming REST services

Are there any examples of a variable being normally distributed that is *not* due to the Central Limit Theorem?

Im going to France and my passport expires June 19th

Why didn't Boeing produce its own regional jet?

Intersection Puzzle

Bullying boss launched a smear campaign and made me unemployable

How much of data wrangling is a data scientist's job?

How to show a landlord what we have in savings?

Examples of smooth manifolds admitting inbetween one and a continuum of complex structures

CAST throwing error when run in stored procedure but not when run as raw query

Size of subfigure fitting its content (tikzpicture)

Is "remove commented out code" correct English?

Why is consensus so controversial in Britain?

Short story with a alien planet, government officials must wear exploding medallions

What exploit Are these user agents trying to use?

Is it acceptable for a professor to tell male students to not think that they are smarter than female students?

What about the virus in 12 Monkeys?

If human space travel is limited by the G force vulnerability, is there a way to counter G forces?

What's the in-universe reasoning behind sorcerers needing material components?

Weak star convergence of Borel probability measures on a metric space

convergence of total variation measureWeak convergence of measure and supportweak-*-convergence of measures ==> convergence of the total mass?Weak convergence in probability and functional analysisAre probability measures weak-* closed?Weak convergence implies convergence on continuous functionsThe Lévy metric metrizes weak convergencevague and weak topology on separable Hilbert spaceJoint convergence of measures and functions: when do integrals converge?Definition of weak convergence of probability measures and weak-* convergence

1

$begingroup$

Let $(X,rho)$ be a compact metric space and let $P(X)$ be the set of Borel probability measures on the Borel $sigma$-algebra of $X$. Suppose $mu_n,mu in P(X)$ for $n in mathbbN$ such that $mu_n to mu$ in the weak star topology, i.e.
$$lim_ntoinftyint_X f(x) hspace1mm dmu_n(x) = int_X f(x) hspace1mm dmu(x)$$
for all continuous functions $f colon X to mathbbC$. If $E subseteq X$ is a Borel set satisfying $mu(E) = mu(textrmInt(E))$, then using the regularity of Borel measures on metric spaces, given $epsilon > 0$ we can produce compact sets $K_1 subseteq textrmInt(E), K_2 subseteq E$ and an open set $U supseteq E$ such that
$$mu(U) - mu(E), mu(E) - mu(K_1),mu(E) - mu(K_2) < epsilon.$$
By Urysohn's lemma there exist continuous functions $f colon X to [0,1]$ and $g colon X to [0,1]$ such that $textrmsupp(f) subseteq U, textrmsupp(g) subseteq textrmInt(E)$ and $f(x) = g(y) = 1$ if $x in K_2, y in K_1$. We have
$$int_X g(x) hspace1mm dmu_n(x) leq mu_n(E) leq int_X f(x) hspace1mm dmu_n(x)$$
for all $n in mathbbN$ by the construction of $f$ and $g$. By the choice of the sets $K_1,K_2,U$ and the weak star convergence of $mu_n_n=1^infty$ it follows that
$$mu(E) - epsilon leq int_X g(x) hspace1mm dmu(x) leq liminf_ntoinftymu_n(E) leq limsup_ntoinftymu_n(E) leq int_X f(x) hspace1mm dmu_n(x) leq mu(E) + epsilon.$$
This shows that $lim_ntoinftymu_n(E)$ exists and is equal to $mu(E)$.

My question is what additional assumptions do we need to conclude that $lim_ntoinftymu_n(E) = mu(E)$ holds for all Borel subsets $E subseteq X$.

real-analysis functional-analysis measure-theory weak-convergence

share|cite|improve this question

edited Mar 21 at 5:38

Ethan Alwaise

asked Mar 21 at 5:16

Ethan AlwaiseEthan Alwaise

6,471717

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I'm confused by your question. You gave an assumption on $E$ that guarantees $lim_n mu_n(E) = mu(E)$, but you end up asking what conditions on $(mu_n)_n$ and $mu$ we need to ensure that $lim_n mu(E_n) = mu(E)$ for all $E$? I'm not exactly sure what kind of answer you are looking for. Also, I think you want $X$ to be compact, so that $int_X gdmu$ always makes sense.
  $endgroup$
  – mathworker21
  Mar 21 at 5:31
  
  

  
  
  
  

add a comment | 

1

$begingroup$

Let $(X,rho)$ be a compact metric space and let $P(X)$ be the set of Borel probability measures on the Borel $sigma$-algebra of $X$. Suppose $mu_n,mu in P(X)$ for $n in mathbbN$ such that $mu_n to mu$ in the weak star topology, i.e.
$$lim_ntoinftyint_X f(x) hspace1mm dmu_n(x) = int_X f(x) hspace1mm dmu(x)$$
for all continuous functions $f colon X to mathbbC$. If $E subseteq X$ is a Borel set satisfying $mu(E) = mu(textrmInt(E))$, then using the regularity of Borel measures on metric spaces, given $epsilon > 0$ we can produce compact sets $K_1 subseteq textrmInt(E), K_2 subseteq E$ and an open set $U supseteq E$ such that
$$mu(U) - mu(E), mu(E) - mu(K_1),mu(E) - mu(K_2) < epsilon.$$
By Urysohn's lemma there exist continuous functions $f colon X to [0,1]$ and $g colon X to [0,1]$ such that $textrmsupp(f) subseteq U, textrmsupp(g) subseteq textrmInt(E)$ and $f(x) = g(y) = 1$ if $x in K_2, y in K_1$. We have
$$int_X g(x) hspace1mm dmu_n(x) leq mu_n(E) leq int_X f(x) hspace1mm dmu_n(x)$$
for all $n in mathbbN$ by the construction of $f$ and $g$. By the choice of the sets $K_1,K_2,U$ and the weak star convergence of $mu_n_n=1^infty$ it follows that
$$mu(E) - epsilon leq int_X g(x) hspace1mm dmu(x) leq liminf_ntoinftymu_n(E) leq limsup_ntoinftymu_n(E) leq int_X f(x) hspace1mm dmu_n(x) leq mu(E) + epsilon.$$
This shows that $lim_ntoinftymu_n(E)$ exists and is equal to $mu(E)$.

My question is what additional assumptions do we need to conclude that $lim_ntoinftymu_n(E) = mu(E)$ holds for all Borel subsets $E subseteq X$.

real-analysis functional-analysis measure-theory weak-convergence

share|cite|improve this question

edited Mar 21 at 5:38

Ethan Alwaise

asked Mar 21 at 5:16

Ethan AlwaiseEthan Alwaise

6,471717

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I'm confused by your question. You gave an assumption on $E$ that guarantees $lim_n mu_n(E) = mu(E)$, but you end up asking what conditions on $(mu_n)_n$ and $mu$ we need to ensure that $lim_n mu(E_n) = mu(E)$ for all $E$? I'm not exactly sure what kind of answer you are looking for. Also, I think you want $X$ to be compact, so that $int_X gdmu$ always makes sense.
  $endgroup$
  – mathworker21
  Mar 21 at 5:31
  
  

  
  
  
  

add a comment | 

$begingroup$

Let $(X,rho)$ be a compact metric space and let $P(X)$ be the set of Borel probability measures on the Borel $sigma$-algebra of $X$. Suppose $mu_n,mu in P(X)$ for $n in mathbbN$ such that $mu_n to mu$ in the weak star topology, i.e.
$$lim_ntoinftyint_X f(x) hspace1mm dmu_n(x) = int_X f(x) hspace1mm dmu(x)$$
for all continuous functions $f colon X to mathbbC$. If $E subseteq X$ is a Borel set satisfying $mu(E) = mu(textrmInt(E))$, then using the regularity of Borel measures on metric spaces, given $epsilon > 0$ we can produce compact sets $K_1 subseteq textrmInt(E), K_2 subseteq E$ and an open set $U supseteq E$ such that
$$mu(U) - mu(E), mu(E) - mu(K_1),mu(E) - mu(K_2) < epsilon.$$
By Urysohn's lemma there exist continuous functions $f colon X to [0,1]$ and $g colon X to [0,1]$ such that $textrmsupp(f) subseteq U, textrmsupp(g) subseteq textrmInt(E)$ and $f(x) = g(y) = 1$ if $x in K_2, y in K_1$. We have
$$int_X g(x) hspace1mm dmu_n(x) leq mu_n(E) leq int_X f(x) hspace1mm dmu_n(x)$$
for all $n in mathbbN$ by the construction of $f$ and $g$. By the choice of the sets $K_1,K_2,U$ and the weak star convergence of $mu_n_n=1^infty$ it follows that
$$mu(E) - epsilon leq int_X g(x) hspace1mm dmu(x) leq liminf_ntoinftymu_n(E) leq limsup_ntoinftymu_n(E) leq int_X f(x) hspace1mm dmu_n(x) leq mu(E) + epsilon.$$
This shows that $lim_ntoinftymu_n(E)$ exists and is equal to $mu(E)$.

My question is what additional assumptions do we need to conclude that $lim_ntoinftymu_n(E) = mu(E)$ holds for all Borel subsets $E subseteq X$.

real-analysis functional-analysis measure-theory weak-convergence

share|cite|improve this question

edited Mar 21 at 5:38

Ethan Alwaise

asked Mar 21 at 5:16

Ethan AlwaiseEthan Alwaise

6,471717

$endgroup$

share|cite|improve this question

edited Mar 21 at 5:38

Ethan Alwaise

asked Mar 21 at 5:16

Ethan AlwaiseEthan Alwaise

6,471717

share|cite|improve this question

edited Mar 21 at 5:38

Ethan Alwaise

asked Mar 21 at 5:16

Ethan AlwaiseEthan Alwaise

6,471717

asked Mar 21 at 5:16

Ethan AlwaiseEthan Alwaise

6,471717

asked Mar 21 at 5:16

Ethan AlwaiseEthan Alwaise

6,471717