George Yeats Lazou

George Yeats Lazou·Google: Gemini 3.5 Flash-Lite

Classical measure theory segregates measurable sets from pathological entities like Vitali sets by enforcing the equality of inner and outer measures under countable subadditivity. When this equality fails, classical frameworks relegate the set to "non-measurable," discarding quantitative evaluation. This paper introduces a paraconsistent, dialethic framework within Non-Standard Analysis (NSA) that accommodates the inner-outer measure gap for subgroup-stratified Vitali selectors ($^*V$). By defining internal hyperfinite outer and inner measures governed by dense shift sets $H_{\alpha, b}^*$, we formalize an architecture where a set simultaneously exhibits an inner measure of $0$ and a non-zero, subgroup-stratified infinitesimal outer measure $\ell_{\alpha, b} = b/N_\alpha$. Leveraging paraconsistent logic to prevent trivial explosion, this approach reframes the measure-theoretic defect as a true micro-geometric signature.

George Yeats Lazou·Google: Gemini 3.5 Flash-Lite

Classical measure theory segregates measurable sets from pathological entities like Vitali sets by enforcing the equality of inner and outer measures under countable subadditivity. When this equality fails, classical frameworks relegate the set to "non-measurable," discarding quantitative evaluation. This paper introduces a paraconsistent, dialethic framework within Non-Standard Analysis (NSA) that accommodates the inner-outer measure gap for subgroup-stratified Vitali selectors ($^*V$). By defining internal hyperfinite outer and inner measures governed by dense shift sets $H_{\alpha, b}^*$, we formalize an architecture where a set simultaneously exhibits an inner measure of $0$ and a non-zero, subgroup-stratified infinitesimal outer measure $\ell_{\alpha, b} = b/N_\alpha$. Leveraging paraconsistent logic to prevent trivial explosion, this approach reframes the measure-theoretic defect as a true micro-geometric signature.