Wednesday, November 6, 2019

Tiny note on an explicit metric on rational lower limit topology $\tilde{\mathbb{R}_{\ell}}$

The lower limit topology on the real line $\mathbb{R}_{\ell}$ is not metrizable. However, $\displaystyle \tilde{\mathbb{R}_{\ell}}$ the rational lower limit topology on $\displaystyle \mathbb{R}$ generated by basis: $\displaystyle [q_1,q_2)$ where, $\displaystyle q_1,q_2 \in \mathbb{Q}$ is metrizable (it's regular + 2nd countable, hence the result follows by Urysohn's metrization theorem). It's interesting to actually write down the metric one gets from Urysohn's theorem.

(Source: Our Prof asked this problem during a Topology lecture in 2016.)

 The key idea is to observe that for any enumeration $\{q_n\}_{n \in \mathbb{N}}$ of the rationals $\mathbb{Q}$, the countable family of continuous function $\displaystyle f_n(x) := \begin{cases} 1 & \text{ if } x \ge q_n \\ 0 & \text{ otherwise }\end{cases}$ (continuous in $\displaystyle \tilde{\mathbb{R}_{\ell}}$ topology) separates disjoint basic clopen neighborhoods.
Hence, the evaluation map $\displaystyle f: \tilde{\mathbb{R}_{\ell}} \to \{0,1\}^{\omega}$ given by $\displaystyle f(x) = (f_n(x))_{n \in \mathbb{N}}$ is an imbedding into the Cantor space $\displaystyle \{0,1\}^{\omega}$ with product topology (as it separates points and closed sets).
Thus the pullback of the metric on Cantor space v.i.a. $f$-imbedding gives us an explicit metric for $\displaystyle \tilde{\mathbb{R}_{\ell}}$: $$d(x,y) = \sum\limits_{n=1}^{\infty} \frac{|f_n(x) - f_n(y)|}{2^n} = \begin{cases} \sum\limits_{x < q_n \le y} \frac{1}{2^n} & \text{ when } x < y \\ 0 & \text{ when } x = y.\end{cases}$$
The function $\displaystyle g(x) = \sum\limits_{q_n \le x} \frac{1}{2^n}$ is monotone increasing and precisely continuous at irrationals and only right continuous at rational points in the standard $\mathbb{R}$ topology as one would expect, and has jumps of magnitude $\displaystyle \frac{1}{2^n}$ at the point $\displaystyle x = q_n$ for each $n \in \mathbb{N}$. The metric can be written as $\displaystyle d(x,y) = |g(x) - g(y)|$.

Update (06.11.2019): I recently came across this interesting post in stack What is a metric for $Q$ in the lower limit topology? where the following interesting metric was proposed for $\mathbb{Q}_\ell$ (the rational lower limit topology):
Write your rationals as "mixed fractions," that is as their integer floor plus a fractional part and define
$$d\left(a\frac{p}{q},b\frac{r}{s}\right)=\begin{cases}|a-b|,& a\neq b\\ d'\left(\frac{p}{q},\frac{r}{s}\right), &\textrm{otherwise} \end{cases}.$$
Define the distance between pure fractions separately. Assume WLOG that $\frac{p}{q}\leq \frac{r}{s}$. Then set $$d'\left(\frac{p}{q},\frac{r}{s}\right)=\max\left(\left|\frac{p}{q}-\frac{r}{s}\right|,\frac{1}{m}\right), \frac{k}{m} \in \left(\frac{p}{q},\frac{r}{s}\right].$$ 
Compared to the jump function $g$ mentioned earlier this metric seems to be modeled on the discontinuities of the Thomae's function. To elaborate this point let us consider the separating family of functions $(f_r)_{r \in \mathbb{Q}}$ to be $$f_r(x) := \begin{cases} 1/q & \text{ if } x \ge r \\ 0 & \text{otherwise} \end{cases}$$ where, $r = p/q \in \mathbb{Q}$ with $(p,q) = 1$. For a fixed $r \in \mathbb{Q}$ the metric $$d_r(x,y) := \begin{cases} |x-y| & \text{ if } x,y \ge r \text{ or } x,y \le r \\ \max\left(|x-y|, 1/q\right) & \text{ if } x < r \le y\end{cases}$$ precisely makes $f_r$ continuous. Modifying the above metric to incorporate jumps at all $r \in \mathbb{Q}$ which has denominator $q$ in irreducible representation followed by a factor $q$ scaling we may consider $$d_q^\prime(x,y) := \begin{cases} q|x-y| & \text{ if } x,y \in \left[\frac{p-1}{q},\frac{p}{q}\right) \text{ for some } p \in \mathbb{Z} \\ \max\left(q|x-y|, 1\right) & \text{ if } x < \frac{p}{q} \le y \end{cases}$$ which now precisely makes those $f_r$'s continuous s.t., denominator of $r \in \mathbb{Q}$ in irreducible representation is $q$. Then the product metric constructed from these $\{d_q'\}_{q \in \mathbb{N}}$ viz. $$d' := \sup_{q \in \mathbb{N}} \left(\frac{\max(d_q',1)}{q}\right)$$ gives a metric compatible to $\mathbb{Q}_{\ell}$ equivalent to the one proposed in the post. Then Thomae's function models the 'jump set' of the metric $d'$.

The underlying "big gun" that would connect both the metrics on $\mathbb{Q}_\ell$ exhibited above would be the following theorem due to Sierpinski:
Theorem (Sierpinski, 1920). A dense in itself countable metric space is homeomorphic to $Q$.
 Both the metrics $d$ and $d'$ above are constructed from explicit homeomorphisms of $\mathbb{Q}_\ell$ with $\mathbb{Q}$.

Thursday, May 25, 2017

Re: A note on a problem from Rudin's Real and Complex Analysis

This post is a short note regarding problem 14 from Page-59 of Rudin's Real and Complex Analysis.

Problem: Let, $f$ be a real-valued Lebesgue measurable function on $\mathbb{R}^k$, prove that there exists Borel functions $g$ and $h$ such that $\mu_k(\{x \in \mathbb{R}^k: g(x) \neq h(x)\}) = 0$ and $g(x) \le f(x) \le h(x)$ for all $x \in \mathbb{R}^k$.  (where, $\mu_k$ is the Lebesgue measure on $\mathbb{R}^k$)

As stated the proposition is incorrect. Best we can do is find Borel functions $g$ and $h$ such that $g(x) = h(x)$ a.e. $[\mu_k]$ and $g(x) \le f(x) \le h(x)$ a.e. $[\mu_k]$.

To see why it's not always possible to 'approximate' a lebesgue measurable function from both above and/or below by Borel functions, we rely on a cardinality argument. Roughly speaking, there aren't enough Borel functions to account for every modification of a lebesgue measurable function in a measure zero set.

Lemma: The cardinality of real Borel measurable functions on $\mathbb{R}$ is $2^{\aleph_0}$.

Proof: Since, constant functions are Borel measurable there are atleast $2^{\aleph_0}$ Borel functions. Again, to determine a Borel function $g$ it suffices to determine the sequence of sets $g^{-1}(q,\infty)$ for each $q \in \mathbb{Q}$ (ordered by inclusion). Since, there are $2^{\aleph_0}$ (i.e., the cardinality of Borel subsets of $\mathbb{R}$)-choices for each $g^{-1}(q,\infty)$ as $q$ varies over $\mathbb{Q}$, there are atmost $\left(2^{\aleph_0}\right)^{\aleph_0} = 2^{\aleph_0}$ real Borel measurable functions on $\mathbb{R}$. $\boxed{}$

Let us index the Borel functions by $\{g_j: j \in J\}$, where, $J$ is some indexing set with cardinality $2^{\aleph_0}$.

Start with the Cantor set $C \subset \mathbb{R}$ which is a set of lebesgue measure zero. Let $C = C_1 \sqcup C_2$ be a partition into subsets such that $|C_1| = |C_2| = 2^{\aleph_0}$. Let us index the elements of $C_1 = \{x_j : j \in J\}$ and $C_2 = \{x_j' : j \in J\}$. Now, if we consider a functions $f: C \to \mathbb{Z}$ such that $f(x_j) < g_j(x_j)$ and $f(x_j') > g_j(x_j')$ for each $j \in J$ (such a $f$ is possible to construct since $\mathbb{Z}$ is an unbounded discrete subset of $\mathbb{R}$), and extend it to $\mathbb{R}$ by defining $f(x) = 0$ for $x \in \mathbb{R}\setminus C$, we have constructed a lebesgue measurable function $f$ which violates the requirement of the proposition form the book.

N.B.: If we were given the scenario that $f$ is a 'bounded' real-lebesgue measurable function on $\mathbb{R}$ (or if, $f:\mathbb{R} \to [-\infty,\infty]$ is lebesgue measurable on the extended real-line), then it is possible to construct borel functions (respectively, extended Borel functions) with the required property that is $g \le f \le h$ for all $x \in \mathbb{R}$. As long as range of $f$ has unbounded discrete subset we may modify $f$ in the above fashion inside an uncountable subset of measure zero such that it is no longer possible to be bounded by Borel functions at all points.