monotonicity of measure

Monotonicity of measure is the obvious-sounding but essential rule that a smaller set cannot have larger size. If one set sits inside another, the inner one’s measure is at most the outer one’s. It is the order-preserving backbone of the whole theory, used so constantly that it is rarely stated aloud.

Formally, if mu is a measure and A is a subset of B with both measurable, then mu(A) is at most mu(B). The proof is one line: write B as the disjoint union of A and B minus A, apply additivity to get mu(B) = mu(A) + mu(B minus A), and observe that mu(B minus A) is non-negative. The same one-line argument works for any non-negative additive set function, including outer measures.

Monotonicity is what lets you bound the measure of a complicated set by sandwiching it between sets you understand, and it underlies the squeeze-style estimates that pervade integration. A caveat worth stating: monotonicity gives mu(B) = mu(A) + mu(B minus A) only when A is contained in B; for general overlapping sets the right tool is subadditivity, mu(A union B) at most mu(A) + mu(B), not equality.