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Apr
23
comment Showing that a set is open in a topological space
Yes, your reasoning is enough. Denote $B_{x}$ as the open neighbourhood $B\subset A$ of $x$. What you need to show is that $A=\cup_{x\in A}B_{x}$.
Apr
23
revised Transport map question
added 1 characters in body
Apr
23
answered Transport map question
Apr
23
comment Finitely additive probabilities in the real line
Also $\nu_{r}$, as defined above, is a constant for each $r$. What type of measure should it represent?
Apr
23
comment Finitely additive probabilities in the real line
What topology do you consider on the set of measures over $\mathbb{R}$? In other words, what do you mean by measures $\lambda_{r}$ converging to the Lebesgue measure? And note that it is an uncountable family of measures, not a sequence, unless you consider $r\in\mathbb{N}$ ofcourse.
Apr
23
comment inequality proof of $x^{y-1} \ge xy$
Induction works for natural numbers, which in this particular inequality is not the case.
Apr
23
revised inequality proof of $x^{y-1} \ge xy$
edited title from y^{y-1} to x^{y-1}
Apr
23
suggested approved edit on inequality proof of $x^{y-1} \ge xy$
Apr
23
comment Ignoring absolute values for integration
@Siminore. But doesn't there exist a concern that $x(t)$ may also take values above $1$, making $ln(1-x)$ undefined?
Apr
23
revised A local base problem
added 2 parenthesis and modified the set notions to make it more clear and to avoid confusions.
Apr
23
suggested approved edit on A local base problem
Apr
23
answered Word problem about elevator capacity of children vs adults.
Apr
23
comment Determine which of the following functions are uniformly continuous on the open unit interval (0,1)
@Siminore. I'm pretty sure that Chandrasekhar is familiar with the definition of a Lipschitz map. Your terminology, in particular the word "lipschitzianity", was likely what caused the confusion.
Apr
23
comment How to move a point onto a line, but leave it alone if it is already on the line?
Could you perhaps rephrase the last two sentences? What do you mean by line always remaining on line and moving line only if it is not on line?
Apr
23
comment Random variable (What is the meaning of a $g(X)$ ?)
Since our interest lies in the $\mu$-measurability of certain sets obtained as preimages by the function, it is quite natural to refer to the structure. It is also a means to avoid repeating the same phrase 'an element of the underlying $\sigma$-algebra $\mathscr{M}$', where $\mathscr{M}$ is indeed the collection of $\mu$-measurable sets. If everything is clear from the context, you may leave the measure out, yet I do argue that it is a quite standard notion to have it there, especially when you have multiple $\sigma$-algebras defined on $\Omega$...
Apr
23
comment Random variable (What is the meaning of a $g(X)$ ?)
Yeah, that would also work for example.
Apr
23
comment Random variable (What is the meaning of a $g(X)$ ?)
Here's small explanation. In this case, $g(X)$ is $\mu$-measurable if the preimage $g(X)^{-1}(U)=X^{-1}(g^{-1}(U))$ is $\mu$-measurable for all open $U\subset \mathbb{R}$. If $g^{-1}(U)$ is open, (which would require continuity), then the measurability of $X$ implies that $g(X)$ is measurable. Unless we know anything about $g$, then we don't know anything about $g^{-1}(U)$.
Apr
23
comment Random variable (What is the meaning of a $g(X)$ ?)
You usually need to consider something from $g$ in order for $g(X)$ to be $\mu$-measurable. If $g$ is continuous, then $g(X)$ will be $\mu$-measurable function aswell, i.e. a random variable.
Apr
23
answered Random variable (What is the meaning of a $g(X)$ ?)
Apr
23
comment Compactness properties imply continuity
Yeah, that's correct.