Zariski's main theorem without inverse limits

pr.chance – Confusion round uniform integrability and Vitali convergence theorem Answer

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pr.chance – Confusion round uniform integrability and Vitali convergence theorem

motivation

The conception of uniform integrability is necessary for the formulation of the Vitali convergence theorem. Unfortunately, totally different authors outline uniform integrability otherwise, which causes fairly a little bit of confusion, resembling from the variety of questions on uniform integrability and the Vitali convergence theorem in. emerges MSE. However, after studying all of those Q&A, I really feel that many college students are quiet confused. In this submit, I need to limpid away all confusion as soon as and for all.

Definitions

Throughout this submit, I might employ the next definitions.

Permit $ (X, mathcal {F}, mu) $ breathe a touchstone area and $ Phi $ a group of measurable features $ (X, mathcal {F}) $, assuming values ​​within the prolonged actual line.

  1. We say $ Phi $ is uniformly restricted in $ L ^ 1 $ if
    $$ sup_ {f in Phi} int_X | f | d mu < infty $$
  2. We say $ Phi $ doesn’t avoid into upright infinity if for any $ epsilon> 0 $, there’s $ M> 0 $ in order that
    $$ sup_ {f in Phi} int_ geq M | f | d mu < epsilon $$
  3. We say $ Phi $ doesn’t avoid to the breadth of infinity if for any $ epsilon> 0 $, there’s $ m> 0 $ in order that
    $$ sup_ {f in Phi} int_ f | f | d mu < epsilon $$
  4. We convene $ Phi $ integrable on the identical time if for any $ epsilon> 0 $, there’s $ delta> 0 $ in order that at any time when $ A in mathcal {F} $ is a measurable quantity, in order that $ mu (A) < delta $, we have now
    $$ sup_ {f in Phi} int_A | f | d mu < epsilon $$
  5. We convene $ Phi $ fastened if for any $ epsilon> 0 $, there’s a measurable quantity $ X_0 in mathcal {F} $ in order that $ mu (X_0) < infty $ and
    $$ sup_ {f in Phi} int_ {X setminus X_0} | f | d mu < epsilon $$

confusion

Here comes the complicated sever about defining unified integrability:

  1. In measurement idea textbooks, uniform integrability is often outlined as being equally integrable.
  2. In chance idea textbooks, uniform integrability is often outlined in such a route that it doesn’t avoid into upright infinity.
  3. In this weblog entry, Professor Tao outlined uniform integrability as uniformly constrained in $ L ^ 1 $, not vertically innumerable and never infinitely vast.

What provides to the confusion is that the varied definitions of unit integrability are solely equal beneath inescapable assumptions, whereas in common they don’t seem to be equal. In addition, totally different authors formulate the Vitali convergence theorem beneath totally different definitions of the uniform integrability.

speculation

It is thought that if $ mu $ is a finite touchstone, then $ Phi $ doesn’t avoid in upright infinity if and solely whether it is uniformly bounded in $ L ^ 1 $ and may breathe built-in on the identical time. For the common illustration, I might love to pose the next assertion. To keep away from confusion, I might keep away from the time period “uniformly integrable” altogether.

Permit $ (X, mathcal {F}, mu) $ breathe a touchstone area and $ Phi $ a group of measurable features $ (X, mathcal {F}) $, assuming values ​​within the prolonged actual line. We do not make some other assumptions about it $ mu $ or $ Phi $. Then the next circumstances are equal:

  1. $ Phi $ is uniformly restricted in $ L ^ 1 $, doesn’t avoid into upright infinity and doesn’t avoid into width infinity
  2. $ Phi $ doesn’t avoid into upright infinity and is dense
  3. $ Phi $ is uniformly restricted in $ L ^ 1 $, equally integrable and taut

Well let $ (f_n) $ breathe a sequence of measurable features and $ f $ one other measurable duty $ (X, mathcal {F}) $. Suppose that:

  1. the gathering $ Phi = {f_n: n in mathbb {N} } $ fulfills one of many above-mentioned equal circumstances,
  2. the sequence $ (f_n) $ converges to $ f $ both virtually all over the place or sparsely,

Then we have now $ f in L ^ 1 ( mu) $, and $ (f_n) $ converges to $ f $ in
$ L ^ 1 ( mu) $.

question

  1. Is my assertion rectify?
  2. Is there a bespeak or publication that’s making concerted efforts to limpid up the confusion about unified integrability and the Vitali convergence theorem?

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