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Doing math.


Apr
23
comment How to make sure that a Point remain on 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.
Apr
23
revised Compactness properties imply continuity
deleted 271 characters in body
Apr
23
answered Compactness properties imply continuity
Apr
23
comment Integral of an $L^2$ function
@NateEldredge. Sure, that would also do it.
Apr
22
revised Absolute value and distance traveled
inputted some latex symboling.
Apr
22
suggested suggested edit on Absolute value and distance traveled
Apr
22
comment Showing Unit sphere is convex
It was edited while ago, but thanks Abdelmajid.
Apr
22
revised Showing Unit sphere is convex
added 4 characters in body
Apr
22
answered Showing Unit sphere is convex
Apr
22
answered What are possible variations of the definition of $\sigma$-additivity?
Apr
22
comment Integral of an $L^2$ function
Maybe one comment worth mentioning is that you can't use the usual $L^{2}(\mathbb{R})$ Cauchy-Schwarz inequality, since the constant function $1$ is not in $L^{2}(\mathbb{R})$. But instead, for each fixed $x,y\in \mathbb{R}$ you may consider $f$ and the constant function $1$ restricted to $[x,y]$ and the Cauchy-Schwarz related to $L^{2}([x,y])$. But anyways, it doesn't really change anything, just a technical detail.
Apr
22
comment Is a Metric space $(X,d)$ with $X=\{x\}$ an open set?
Where in the definition does it require an $y\in X$ so that $d(x,y)=r$, $r>0$? In this case any open ball centered at $x$ would equal the singleton $\{x\}$.
Apr
22
revised Basic proof that $T$ is a topology.
In the question, there was a mistake in the set-notation: /in instead of /subset
Apr
22
suggested suggested edit on Basic proof that $T$ is a topology.