Are right continuous functions measurable?

Question

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.

Answer 1

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.

Answer 2

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.