Lebesgue measure is integral to probability theory, but it has limitations, notably that not all sets are measurable. This raises questions about its applicability in physics, where relevant sets are typically measurable. The discussion touches on the axiom of choice and its implications for mathematical physics, particularly regarding non-measurable sets. The Banach-Tarski paradox illustrates the tension between mathematical theory and physical intuition, suggesting that non-measurable sets could lead to paradoxes if applied to physical objects. Ultimately, the consensus is that measurable sets align better with the properties of physically measurable phenomena.