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Jun
10
comment convergence uniformly of a sequence of functions
I edited my answer according to the newly edited question, and as far as I'm concerned, the current claim does not hold.
Jun
10
revised convergence uniformly of a sequence of functions
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Jun
10
comment Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?
@StefanWalter: I don't.
Jun
10
comment Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?
@StefanWalter: To get back to the topic: each individual number on the real line has probability zero when throwing a dart, yet you always hit one of them. Once you did hit one, an event with probability zero occured. Being likely to happen and being possible to happen are two different things.
Jun
10
revised need help with real analysis question
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Jun
10
comment convergence uniformly of a sequence of functions
I would suggest first proving that $g$ is continuous as a uniform limit of continuous functions. Use this to work around with the hint I gave below. In your comments, is $f(x)$ same as $g(x)$?
Jun
9
answered convergence uniformly of a sequence of functions
Jun
9
comment convergence uniformly of a sequence of functions
Is this a homework? If yes, what have you tried so far?
Jun
9
comment To define a measure, is it sufficient to define how to integrate continuous function?
The part of the proof which shows uniqueness of $\mu$ is nice and fits the OP's question very well: it shows that if $\int f\,d\mu=\int f\,d\nu$ for all $f\in C_{c}(X)$ then $\mu=\nu$. It's done basicly with Urysohn's lemma, and you also need regularity of $\mu,\nu$ or Dynkin's theorem in the proof, which Rudin does not specify explicitly. He just says it is enough to deal closed sets which, however, isn't a trivial remark.
Jun
9
comment To define a measure, is it sufficient to define how to integrate continuous function?
Yeah; there's no doubt that Dynkin is a powerful tool :-) You could, however, also prove it manually by showing that the collection of sets for which such approximation holds true is a $\sigma$-algebra that contains Borel sets. But as you said, Dynkin probably does the job easier.
Jun
9
comment To define a measure, is it sufficient to define how to integrate continuous function?
An alternative way to conclude the proposition's proof is by regularity of finite Borel measures on metric spaces: the measure of Borel sets can be approximated by closed sets from below, so if $\mu=\nu$ on closed sets then they agree on all Borel sets. But Dynkin is ofcourse also very neat.
Jun
9
comment Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?
@StefanWalter: I only told how it works in the $\sigma$-algebra generated by this random variable: I didn't assign anything to them myself. No matter how much we discuss about it, an event having probability zero will not imply it is impossible. To conclude to the last example, lets say we have the option of wind blowing the coin away with probability zero. If you're inside the house, then it's impossible (because there is no wind!) but outside, eventhough it's zero probability, the wind exists and it not impossible. It is only almost sure (a.s.) that the wind does not blow the coin away.
Jun
8
awarded  Constituent
Jun
8
awarded  Caucus
Jun
8
comment What does the following statement mean?
@Ananda: for any sequence $(a_{n})_{n=1}^{\infty}$ of real numbers we have $\limsup\, a_{n}\geq \liminf\, a_{n}$, so I just applied this to the above line.
Jun
8
comment Vectors - Define a vector of length 1 orthogonal to $\vec{v} = (-4 \qquad 3)^t$
What are all the squares in upper corners of each vector symbol?
Jun
8
revised What does the following statement mean?
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Jun
8
revised What does the following statement mean?
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Jun
8
revised What does the following statement mean?
added 1436 characters in body
Jun
8
answered What does the following statement mean?