Can the identity function be expressed as a countable-linear combination of indicator functions? > **Question.** Can the function $$\mathrm{id} : \mathbb{R} \rightarrow \mathbb{R}, \qquad \mathrm{id}_\mathbb{R}(x) = x$$ > > be expressed as a countable-linear combination of indicator functions of subsets of $\mathbb{R}$? _Remark._ One idea for constructing such a thing is to try to find a function $a : \mathbb{Q} \rightarrow \mathbb{R}$ such that $$(\forall x \in \mathbb{R}) \qquad x = \sum_{q \in \mathbb{Q}}a_q[x<q],$$ where the square brackets connote the Iverson bracket. But it's not really clear how to choose the $a_q$'s. Honestly, I think the answer is probably "no." Ideas, anyone? I'm also interested in the (simpler?) problem where $\mathbb{R}$ is replaced by $\mathbb{R}_{\geq 0}$.

Fix for each $0\leq x<1$ some binary representation (for example, say that you use for each $x$ the representation not containing an infinite tail of ones).

Then $$ x=\sum_{j=1}^{\infty}2^{-j} [j\text{th digit in binary representation of }x\text{ is }1] $$ for any $0\leq x<1$, as follows pretty much directly from the definition of binary representations.

To extend this to all of $\mathbb{R}_{\geq}$ just extend the sum to $-\infty$.

To extend it to all of $\mathbb{R}$ just add the same representation with replacements $x\mapsto -x$ and $2^{-j}\mapsto -2^{-j}$