Real analysis & measure theory
Lebesgue's rigorous theory of measure and integration.
Founded: 1902 — Lebesgue's thesis on integration
The claim
No credible dismissal found. Real controversy existed (Borel's constructivist objections, du Bois-Reymond's rigor arguments) but concerned legitimacy, not usefulness.
What happened
Measure-theoretic probability (1909) — Borel's strong law for coin tossing brings countable additivity into probability. Documented in Shafer and Vovk, The Sources of Kolmogorov's Grundbegriffe, Statistical Science 21 (2006).
Axiomatic probability theory (1933) — Kolmogorov's Grundbegriffe — the synthesis that became the standard measure-theoretic foundation of probability.
7 years — founding (1902) to Borel's strong law (1909)
Verdict: Never dismissed, applied anyway.
Confidence: medium-high.