# A conjecture about measurable functions [duplicate]

(Modified to be specific enough so that the other post does not answer my question) I have the following conjecture about measurable functions.

Let $$\Omega$$ be the sample space equipped with $$\sigma$$-algebra $$\mathcal{F}$$, and let $$\mathcal{H} \subseteq \mathcal{F}$$ be a sub-$$\sigma$$-algebra. For each $$\omega \in \Omega$$ we can associate a set that consists of answers to the questions "is $$\omega \in A$$?" for each $$A \in \mathcal{H}$$. Let this be called the $$\mathcal{H}$$-set of $$\omega$$. Concretely, the $$\mathcal{H}$$-set for $$\omega$$ is given by $$\lbrace(A, 1_A(\omega)):A \in \mathcal{H}\rbrace$$. Then, a function $$X: \Omega \to \mathbb{R}$$ is $$\mathcal{H}$$-measurable if and only if $$X$$ is constant across $$\omega$$'s that have the same $$\mathcal{H}$$-set.

I was able to prove one direction of this conjecture, namely that if $$X$$ is $$\mathcal{H}$$-measurable, then it is constant across $$\omega$$'s that have the same $$\mathcal{H}$$-set. However, I do not know how to prove or disprove the other direction -- that if $$X$$ is constant across $$\omega$$'s that have the same $$\mathcal{H}$$-set, then $$X$$ is $$\mathcal{H}$$-measurable.

• Apart from being a duplicate question it is totally unclear what you mean by "knowing whether each $A \in \mathcal{H}$ occurs uniquely". Commented May 17 at 0:46
• @KurtG. The linked post does not answer my question; in particular, I am still uncertain as to whether my conjecture is true, and I do not have a proof. Regarding your second comment, I'm not sure exactly what is unclear about it. Are you familiar with the meaning of "knowing x uniquely determines y"? Commented May 17 at 1:12
• After reading Robert Israel's answer I understand that conjecture now. There is an even simpler counter example: $1_A$ and $2\cdot 1_A$ are both measurable w.r.t. the $\sigma$-algebra $\{\emptyset,A,A^c,\Omega\}\,.$ Knowing whether $A$ occurs obviously not determines them uniquely. Commented May 17 at 4:35
• Restricting ourselves to simple functions $X$ that have only finitely many values and are linear combinations of finitely many indicator functions of disjoint sets $A_i$ one can say that the $\sigma(A_1,\dots,A_n)$-measurability of $X$ is equivalent to knowing from the value of $X$ which $A_i$ has occurred. The converse is your conjecture and not true. Commented May 17 at 6:13
• You are correct. My conjecture was not capturing the intuition I meant to convey. Please see the corrected version above. Commented May 18 at 0:18

Your conjecture is false. Let $$\mathcal H$$ be the $$\sigma$$-algebra of Lebesgue measurable sets in $$\mathbb R$$. Knowing whether each half-line $$(-\infty, x)$$ occurs, i.e. whether $$X < x$$, uniquely determines $$X$$. But there are Lebesgue non-measurable functions $$X$$.