Real Analysis Qualifier at Texas A&M University - College Station | Flashcards & Summaries

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TESTE DEIN WISSEN
Define first countable. 
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TESTE DEIN WISSEN
A set is first countable if every point has a countable neighborhood base.  
Lösung ausblenden
TESTE DEIN WISSEN
Zorn’s Lemma



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TESTE DEIN WISSEN
If X is a partially ordered set and every linearly ordered subset (or chain) of X has an upper bound, then X has a maximal element. 
Lösung ausblenden
TESTE DEIN WISSEN
Define what it means for a function to be measurable. 
Lösung anzeigen
TESTE DEIN WISSEN
A function is measurable if the pull back of every measurable set is measurable. 
Lösung ausblenden
TESTE DEIN WISSEN
If E belongs to the product sigma algebra, do the the x and y sections belong to their respective sigma algebras?
Lösung anzeigen
TESTE DEIN WISSEN
Yes. 
Lösung ausblenden
TESTE DEIN WISSEN
What is a Hausdorff space?
Lösung anzeigen
TESTE DEIN WISSEN
It is a space such that every two distinct points can be enclosed by disjoint open sets. 
Lösung ausblenden
TESTE DEIN WISSEN
What does it means for a set to be normal?
Lösung anzeigen
TESTE DEIN WISSEN
A set is normal if for any two closed sets, you can enclose them in two disjoint open sets. 
Lösung ausblenden
TESTE DEIN WISSEN
Give an example of a sequence of functions that converges uniformly, in measure, but not in L1. 
Lösung anzeigen
TESTE DEIN WISSEN
F_n=(1/n)X_(o, n)
Lösung ausblenden
TESTE DEIN WISSEN
If a sigma algebra is generated a collection of sets, and the pullback of each of the generating sets is measurable, is f measurable?
Lösung anzeigen
TESTE DEIN WISSEN
Yes. 
Lösung ausblenden
TESTE DEIN WISSEN
Suppose a sequence of functions converges in L1. What can be said about it’s pointwise convergence?
Lösung anzeigen
TESTE DEIN WISSEN
There exists a subsequence that converges pointwise  
Lösung ausblenden
TESTE DEIN WISSEN
Define a measure.
Lösung anzeigen
TESTE DEIN WISSEN
A measure is a function from a Sigma algebra such that the empty set has measure zero and has countable additivity. 
Lösung ausblenden
TESTE DEIN WISSEN
Define what a “rectangle“ is in a product measure space. 
Lösung anzeigen
TESTE DEIN WISSEN
A rectangle is the Cartesian product of two measurable sets. 
Lösung ausblenden
TESTE DEIN WISSEN
State Tychonoffs Compactness theorem. 
Lösung anzeigen
TESTE DEIN WISSEN
Product of compact spaces is compact. 
Lösung ausblenden
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Q:
Define first countable. 
A:
A set is first countable if every point has a countable neighborhood base.  
Q:
Zorn’s Lemma



A:
If X is a partially ordered set and every linearly ordered subset (or chain) of X has an upper bound, then X has a maximal element. 
Q:
Define what it means for a function to be measurable. 
A:
A function is measurable if the pull back of every measurable set is measurable. 
Q:
If E belongs to the product sigma algebra, do the the x and y sections belong to their respective sigma algebras?
A:
Yes. 
Q:
What is a Hausdorff space?
A:
It is a space such that every two distinct points can be enclosed by disjoint open sets. 
Mehr Karteikarten anzeigen
Q:
What does it means for a set to be normal?
A:
A set is normal if for any two closed sets, you can enclose them in two disjoint open sets. 
Q:
Give an example of a sequence of functions that converges uniformly, in measure, but not in L1. 
A:
F_n=(1/n)X_(o, n)
Q:
If a sigma algebra is generated a collection of sets, and the pullback of each of the generating sets is measurable, is f measurable?
A:
Yes. 
Q:
Suppose a sequence of functions converges in L1. What can be said about it’s pointwise convergence?
A:
There exists a subsequence that converges pointwise  
Q:
Define a measure.
A:
A measure is a function from a Sigma algebra such that the empty set has measure zero and has countable additivity. 
Q:
Define what a “rectangle“ is in a product measure space. 
A:
A rectangle is the Cartesian product of two measurable sets. 
Q:
State Tychonoffs Compactness theorem. 
A:
Product of compact spaces is compact. 
Real Analysis Qualifier

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