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14h
comment Example of Riemann integrable $f: [0,1] \to \mathbb R $ whose set of discontinuity points is an uncountable and dense set in $[0,1]$
It certainly is!
14h
comment Countable collection of countable sets and Axiom of choice
possible duplicate of The cardinality of a countable union of countable sets, without the axiom of choice
14h
comment Example of Riemann integrable $f: [0,1] \to \mathbb R $ whose set of discontinuity points is an uncountable and dense set in $[0,1]$
In fact, it's discontinuous at all the rationals also. So, the set of discontinuity points is all of [0,1].
1d
comment Complex orthogonal polynomial
I see. So you want $P_n$ to have degree $n$, and maybe you want the inner product to be of the form $\langle f,g \rangle = \int fg\,d\mu$ for a positive measure $\mu$. I was thinking of a completely arbitrary inner product.
2d
answered Showing that a countable product of unit intervals is not compact in the box and uniform topology.
2d
reviewed Leave Closed Computing integration
2d
reviewed Looks OK Hexadecimal Representation
2d
comment Complex orthogonal polynomial
I don't see why the zeros of the polynomials need to be simple, even in the real case. Can't you let $P_1$ be any polynomial with repeated zeros, and use Gram-Schmidt to construct $P_2, P_3, \dots$ orthogonal to $P_1$ and each other?
2d
comment Can a Set Have Infinitely-Many Non-Homeomorphic Topologies?
possible duplicate of Are there uncountably many non homeomorphic ways to topologize a countably infinite set?
2d
comment local martingale bounded below by a DL process
No, I haven't seen that specific statement before, but in light of my (now corrected) proof below, I believe it :-)
2d
revised local martingale bounded below by a DL process
fix proof
2d
comment local martingale bounded below by a DL process
@Richard: Yes, I actually just thought of the same thing. I will update my answer.
2d
comment local martingale bounded below by a DL process
What's your textbook, and where does it seem to use this fact? I am wondering if there are other conditions on $X_t$ that may be relevant.
2d
answered local martingale bounded below by a DL process
Nov
23
comment baire category and the union of dyadic balls of rational center
For example, we might have an enumeration such that $r_0 = 3$, $r_2 = 3.1$, $r_4 = 3.14$, $r_6=3.141$, etc, and the odd-numbered $r_n$ enumerate the rest of the rationals. Then $|\pi - r_{2n}| \le 10^{-n} < 4^{-n} = 2^{-2n}$, so $\pi \in \bigcap_{k=0}^\infty U_k$.
Nov
23
comment baire category and the union of dyadic balls of rational center
But you can't guarantee $\bigcap_{k=0}^\infty U_k = \emptyset$. Depending on the enumeration $\{r_n\}$, there certainly could be an $x$ such that $|x - r_n| < 2^{-n}$ for infinitely many $n$. Such an $x$ would be in $\bigcap_{k=0}^\infty U_k$.
Nov
23
comment baire category and the union of dyadic balls of rational center
I may be a bit dense myself, but I'm not seeing how you conclude that some $\bigcup_{n=k}^\infty B(r_n, 2^{-n})$ is not dense.
Nov
23
revised baire category and the union of dyadic balls of rational center
edited tags
Nov
23
reviewed Leave Closed how likely is it that 2 strangers in a city of 8 million will cross paths?
Nov
23
reviewed Approve suggested edit on Find $E(X)$ and $Var(X)$