Thomas E.
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 Jun8 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. Jun8 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. Jun8 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 $B$. Jun7 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. Jun7 comment Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$? @QiaochuYuan: what sum? Jun7 comment Weak a.s. convergence VS a.s.weak convergence @NateEldredge. thanks for clearing this:) Jun7 comment Weak a.s. convergence VS a.s.weak convergence What measure is $\mathbb{P}$? Jun7 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. Jun7 comment Unit speed geodesics @WillieWong: Thanks. Jun7 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? Jun6 comment Unit speed geodesics Thanks. I knew there existed a simple way to see this!:) Jun6 comment Unit speed geodesics @Thomas: You're probably right. If $d(x,y)=\inf_{\gamma\in\mathscr{D}(x,y)}L_{d}(\gamma)$ for all $x,y\in X$: where $L_{d}$ is the length-function induced by $d$ and $\mathscr{D}(x,y)$ is the collection of those $\gamma\in X^{[0,1]}$ that are continuous, $\gamma(0)=x$ and $\gamma(1)=y$ and $L_{d}(\gamma)<\infty$ - we say that $(X,d)$ is a length space. In other words, if the distance of any pair of points is obtained as infimum over the lengths of rectifiable paths that join the points. Jun6 accepted Unit speed geodesics Jun6 revised Unit speed geodesics added tag Jun6 accepted Compactly supported function? Jun6 asked Unit speed geodesics Jun5 awarded Civic Duty Jun5 comment Cross product of paths is a path with length? By saying that $a(t)$ is a path with length in $[c,d]$, do you mean that the length of $a(t)$ is a real between $c$ and $d$? Jun5 comment Is Koch snowflake a continuous curve? If the convergence is uniform, then a limit of continuous functions is continuous. Jun5 comment (Un-)Countable union of open sets @DaveL.Renfro. True, I didn't somehow even think about that :-) So $T_{1}$ would be the least requirement for this construction.