Say that $C$ is a SSLLN class of subsets of some Polish space $V$ provided that for every sequence of Borel i.i.d.r.v.s $X_1,X_2,...$ with values in $V$, we almost surely have: For every $A$ in $C$, $\frac1n \sum_{k=1}^n 1_A(X_k) \to P(X_1\in A)$. (The important thing is the order of quantifiers....

Motivation: This question is inspired by a talk from Avi Wigderson given on Randomness, where the idea that the randomness is in the eye of the observer is suggested. In the study of information theory and its applications, an intriguing paradox emerges when considering Kolmogorov complexity and ...

Is there any field of mathematics that deals with the role of the observer? E.g., some formulation in which a set is changed, in some unspecified way, when it is observed? Or maybe some philosophy of mathematics that addresses this? (I am more interested in formal mathematical systems than philos...

$\newcommand{\HH}{\mathrm{HH}}$Consider a general spacetime containing an observer, and let $\mathcal{A}_{\mathrm{obs}}$ denote the algebra of observables available to the observer. It has been proposed that the Hartle-Hawking "no boundary" state $\Psi_{\HH}$ of the universe can be interpreted as...