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Are right-continuous function from $\mathbb{R}^n$ to $\mathbb{R}$ necessarily semi-continuous? If not, are they necessarily Borel measurable? Is there a topological characterization of right-continuous functions (as there is of continuous ones)? Are CDFs of $n$-dimensional random vectors measurable?

Note: A function $f: \mathbb{R}^n \longrightarrow \mathbb{R}$ is right-continuous iff it is right-continuous at every point $x \in \mathbb{R}^n$. A function $f: \mathbb{R}^n \longrightarrow \mathbb{R}$ is right-continuous at $x \in \mathbb{R}^n$ iff given any infinite sequence of points in $\mathbb{R}^n$ $(y_0,y_1,\dots)$ that converges to $x$ from above (i.e. the sequence converges to $x$ in the usual, Euclidean sense and in addition every sequence element is greater than or equal to $x$ component-wise), the sequence $(f(y_0), f(y_1), \dots)$ converges to $f(x)$ in the usual sense.

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    You might want to add the definition of right-continuity for a function defined on $\mathbb R^n$ when $n\gt1$.2012-08-02
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    What are the definition of right-continuous function, and semi-continuous?2012-08-02

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The answer to the first question is no, even in the case $n=1$: The characteristic function of the half open interval $[0,1)$ is right continuous, but neither upper nor lower semicontinuous.

A right continuous function $\mathbb{R}\to\mathbb{R}$ is indeed Borel measurable. By definition, the inverse image $E$ of an open set has the property that for any $x\in E$, there is some $\delta>0$ so that $[x‚x+\delta)\subseteq E$. It follows that $E$ is a countable union of half open intervals, and hence is Borel measurable. I am not sure about the answer to this one when $n>1$ (the countable union argument no longer holds), but my guess is that right continuous functions are still measurable.

Topological characterization: If we write $\le$ for pointwise comparison on $\mathbb{R}^n$, we can make a one-sided topology by declaring a set $V\subseteq\mathbb{R}^n$ to be open if, for each $x\in V$, there is some $\delta>0$ so that $\{y\ge x\colon\lvert y-x\rvert<\delta\}\subseteq V$. Then the right continuous maps $\mathbb{R}^n\to\mathbb{R}$ are just the ones that are continuous from this topology to the usual topology on $\mathbb{R}$.

I am not too sure on the CDF question either. (I assume CDF stands for cumulative distribution function, in the sense of $F(x)=\mathrm{P}\{X\le x\}$, where $X$ is a random $n$-vector.) It might help that $F$ is not only right continuous, but also monotone. So the set $\{x\colon F(x)\le p\}$ has a particularly simple structure; I imagine it must be measurable, but right now I don't see a proof.

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    Thanks, Harald, but it is in fact the case n>1 that i am interested in. See my clarification of the term "right-continuous" in a comment to my original post. Also, of the four questions i posed in my original post i am most interested in the last one concerning CDF.2012-08-02
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    @EvanAad: Okay. I'll let my answer stand, though, since at least it provides a simple, partial answer which may be of use. I even expanded on it, though it is still far from a satisfactory answer.2012-08-02
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    Thank you, Harald. Despite your reservations, i think you did actually answer my question re CDFs (and yes, i meant "CDF" as in "Cumulative Distribution Function"), since the one-sided topology you defined is contained in (in fact, generates) the Borel field on R^n (at least so it appears to me, at a cursory glance).2012-08-02
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    In fact, Harald - and please correct me if i'm wrong - the one-sided topology you described can be generated by sets that are the result of intersecting a "generalized ray" of the form [x, infty) (x in R^n) with an open set in R^n.2012-08-02
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    @EvanAad: This topology appears to be the product of $n$ copies of the [Sorgenfrey line](http://en.wikipedia.org/wiki/Sorgenfrey_line). But I don't quite see why the open sets of this topology are Borel. It certainly has a basis of Borel sets, but I don't think it's second countable, so why does it follow that any union of such sets is also Borel?2012-08-02
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    @NateEldredge: Right you are. Thanks for pointing this out. So the question of CDF measurability is still open.2012-08-03
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    @EvanAad: In fact, let $E$ be a non-measurable subset of $\mathbb{R}$ and let $A = \bigcup_{x \in E} [x,\infty)^2$. $A$ is open in our topology on $\mathbb{R}^2$ (the Sorgenfrey plane), but not Borel with respect to the usual topology, since its intersection with the diagonal line $y=-x$ is a skewed copy of $E$. One might think one could exploit this to make a right-continuous function which is not Borel, but I still don't see how.2012-08-03
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    @NateEldredge: Did you mean $\bigcup_{x\in E}[x,\infty)\times[-x,\infty)$?2012-08-03
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    @HaraldHanche-Olsen: Oops, yes, I did.2012-08-03
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    @NateEldredge: Thanks for your input! I'm pleased to inform the CDF question has been resolved in [another thread](http://math.stackexchange.com/questions/178284/are-cumulative-distribution-functions-measurable), so once a counterexample is produced demonstrating a right-continuous function that is not Borel-measurable, i'll be able to mark this thread as "Answered."2012-08-04
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    I'm caving in and marking this thread as "answered", even though the question of whether a $\mathbb{R}^n\rightarrow\mathbb{R}$ right-continuous function can fail to be Borel-measurable has not yet been resolved for $2\leq n$. What i was really after in starting this thread was the part concerning the CDF and this has been resolved in another thread, as mentioned above.2012-09-04
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    Sorry to dig up an old answers but can you please explain why the union is countable? It would seem to me that $\cup_x [x+\delta(x))$ is uncountable.2018-01-02
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Here is a proof that any function $F: \mathbb{R}^n \to \mathbb{R}$ which is right continuous is Borel measurable.

Define $F_n(x) = F\left(\frac{[nx]+1}{n}\right)$, where $[x]$ is the greatest integer smaller than $x$. Clearly $F_n$ are measurable since they are step functions.

From the right continuity of $F$ we get that $F_n\to F$, since $\frac{[nx]+1}{n} \downarrow x$.

Hence since $F$ is the pointwise limit of measurable functions, it is measurable too.

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    Very Nice, Milind!2017-08-24
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    to make this a bit more solid, you would need to clarify that $n=1$ or properly define $[x]$ for an $n$-vector $x$.2018-10-12