Juxtaposition Measure and the Resolution of Classical Non-Measurable Sets
Year: 2026
Type: preprint
Abstract: We introduce \emph{juxtaposition addition} ($\uplus$): identical elements placed side by side yield $0\uplus0\uplus\cdots=1$, corresponding to $0\times\infty=1$ (1D) and $0_s\times\infty_s=1_s$ ($s$-dimensional). Under this regime, Carath\'eodory's condition holds identically for all sets---no non-measurable sets exist. Seven classical non-measurable constructions (Vitali, Bernstein, Sierpi\'nski, Hamel, Ulam, Banach--Tarski, Hausdorff) are resolved with unique phase-proportion assignments determined by construction rules. The Cantor set has 1D juxtaposition measure $2/3$ and $s$-dimensional measure $1_s$, aligning with Hausdorff normalization. Arithmetic operations and the four-step calculus procedure are unified as expansion/contraction duality repeated across the two regimes. Classical measure theory is not overthrown but completed at the microscopic level.
Author: Lu, Sheng
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203880894
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