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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?
added 77 characters in body
Jun
8
revised What does the following statement mean?
added 1436 characters in body
Jun
8
revised What does the following statement mean?
added 1436 characters in body
Jun
8
answered What does the following statement mean?
Jun
8
comment Every path has a simple “subpath”
Can the length of $f$ be infinite? If not, then such $g$ is found with very simple reparameterization argument.
Jun
8
comment Set {1,1} = Set {1}, origin of this convention
If $A=\{1,1,2,3\}$ and $B=\{1,2,3\}$, then $A\subset B$ (any element you take from $A$ is in $B$) and $B\subset A$ implies $A=B$. A good book that I could recommend for reading for related topics is Paul Halmos; Naive Set Theory. It starts building the axioms of set theory very nicely.
Jun
8
comment Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?
@StefanWalter: This doesn't really change anything. Either $C$ is an event of neither tails nor heads (e.g. wind blew the coin away) in which case, again, if $C$ is zero measurable it does not mean that it can not happen. It only says that tails or heads are gotten almost surely. On the hand, if you consider no other options than tail or heads, then $C$ is empty. For the r.v. that takes two values, any set not containing them has empty preimage. In fair coin tossing there are no non-empty zero measurable sets separating $A$ and $B$.
Jun
7
comment Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?
@StefanWalter: It really depends on how you define the word impossible. If impossible means that it can not happen, then clearly this does not coincide with the concept of probability zero and it would indeed be wrong/misguided to say that it would.
Jun
7
comment Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$?
@QiaochuYuan: what sum?
Jun
7
comment Weak a.s. convergence VS a.s.weak convergence
@NateEldredge. thanks for clearing this:)
Jun
7
comment Weak a.s. convergence VS a.s.weak convergence
What measure is $\mathbb{P}$?
Jun
7
comment Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?
Hitting any individual number has probability zero. So with this logic we will never hit any point when throwing a dart. Can you see the flaw in this? Eventhough rationals cumulate zero mass on the real line, this doesn't mean that they don't exist.
Jun
7
comment Unit speed geodesics
@WillieWong: Thanks.
Jun
7
comment Condition on function $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $(a,b)\mapsto | f(a) - f(b)|$ generates a metric on $\mathbb{R}$
@copper.hat: You're probably assuming that $f$ is in addition continuous?
Jun
6
comment Unit speed geodesics
Thanks. I knew there existed a simple way to see this!:)