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
Axiomatic probability theory (1933) — Kolmogorov's Grundbegriffe — the now-universal measure-theoretic foundation of probability.
31 years — founding (1902) to Kolmogorov's axiomatization (1933)
Verdict: Never dismissed, applied anyway.
Confidence: medium-high.