How does one find the expected value of the Conway base-13 function $f:\mathbb{R}\to\mathbb{R}$, when restricted to $f:[0,1]\to[0,1]$?

Question

Main Question:

Using the Lebesgue measure, what is the expected value of the Conway base-13 function $f:\mathbb{R}\to\mathbb{R}$, when restricted to $f:[0,1]\to[0,1]$?

Attempt:

I’m not sure what the answer is; infact, I’m not sure an answer exists.

According to this, the Conway base-13 function is not computable (see Noah Schweber’s comment for a definition on computability).

Moreover, I find the function difficult to understand. (I only have knowledge up to Intro to Advanced Mathematics).

Due to my voracious interest in math, I’m unable to withhold questions until I’ve gained more knowledge. Does the main question have a positive answer?

Edit: Turns out the Conway base-13 function cannot have a range of $[0,1]$ and doesn’t satisfy the main question of this post. I will leave the question as it is.

This function does not look measurable. Hence, its expected value should be undefined. — Kovomaka, May 30 '23 at 19:58
@Kovomaka Then Noah Schweber and I were wrong about the function satisfying the main question of this post — Arbuja, May 30 '23 at 20:02

score 3 · Accepted Answer · answered May 30 '23 at 20:07

3

In what way do you plan on restricting $f : \mathbb R \to \mathbb R$ to $[0,1] \to [0,1]$? For any $r \in \mathbb R$, there exists a $c \in [0,1]$ where $f(c)=r$. So the only way of turning $f$ into a function $[0,1] \to [0,1]$ would be to compose it with some map that shrinks the image from $\mathbb R$ to $[0,1]$ -- Say something like $x \mapsto x - \lfloor x \rfloor$.

So, it is nonsensical to ask about any property of "the" restriction of $f$ into $[0,1] \to [0,1]$, since no natural and canoncial choice exists to transform one into the other. Perhaps you meant $[0,1] \to \mathbb R$?

answered May 30 '23 at 20:07

Snared

865

Can you check if the Conway base-13 function satisfies the main question of this post. If not, the Noah Schwebers comments were incorrect. – Arbuja May 30 '23 at 20:12
1

It obviously doesn't - $f$ surjects onto $\mathbb R$ on any interval $(a,b) \subset \mathbb R$ – Snared May 30 '23 at 20:15
Does the main question to this post have a function that satisfies all the listed criteria? – Arbuja May 30 '23 at 20:18
3

I only intend to answer this question, not any new ones you've linked in the comments. – Snared May 30 '23 at 20:20

How does one find the expected value of the Conway base-13 function $f:\mathbb{R}\to\mathbb{R}$, when restricted to $f:[0,1]\to[0,1]$?

1 Answers1

Linked