What is the span of the Haar system in the $L^\infty$ norm?

Question

The Haar system in $[0,1]$ has a closed span in the $L^\infty[0,1]$ norm which contains all the continuous functions in $[0,1]$. In fact, it contains all the piecewise continuous functions with possibly finite discontinuities at dyadic numbers. Is there a precise characterization of the closed span, as a closed subspace of $L^\infty$?

Answer 1

If I follow the notation of https://www.encyclopediaofmath.org/index.php/Haar_system then, ignoring a finite number of points (a null set) then the first $2^n$ Haar functions has span equal to functions which are piecewise constant on each "dyadic" interval of length $2^{-n}$, to be precise, the intervals $(2^{-n}k, 2^{-n}(k+1))$ for $0\leq k<2^n$.

I think it thus follows that $f\in L^\infty$ is in the norm closed linear span if for each $\epsilon>0$, there is some $n$ such that $f$ varies by less than $\epsilon$ on each dyadic interval of length $2^{-n}$, all up to a null set.

I find it hard to give a more precise characterisation. A heuristic for why this is so is to see that $L^\infty[0,1]$ does not depend on the topology of $[0,1]$. We could consider any Standard probability space.

In particular, lets consider the most standard, $\{-1,1\}^\mathbb N$. We can think of the Haar system here as being the constant function $\psi_{0,0}\equiv 1$, and the functions $\psi_{n,k}$ defined by, on sequences $x=(x_i)\in\{-1,1\}^\mathbb N$, $$ \psi_{1,0}(x) = x_1, $$ $$ \psi_{2,0}(x) = x_2 \delta_{x_1,-1}, \quad \psi_{2,1}(x) = x_2 \delta_{x_1,1} $$ and so forth (probably with some re-scaling).

We can also think of $\{-1,1\}^\mathbb N$ as the Cantor space, and then the span of the Haar functions agrees with the span of the indicator functions of the basic clopen sets in $\{-1,1\}^\mathbb N$ (i.e. the sets formed by all sequences with some common prefix). All such functions are continuous, and conversely, any continuous function can be approximated by a span of such functions. (I see this most easily by turning $\{-1,1\}^\mathbb N$ into a metric space by $d(x,y)=1/k$ where $k$ is the least index where x and y differ.)

So, the span of the Haar system in $L^\infty$ of the Cantor space is all continuous functions. The same applies to the Cantor set of course. But this is all slightly cheating by redefining the question.