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We apply this to design expected-utility-optimal e-values for group invariance, which include both Neyman\u2013Pearson-optimal tests and log-optimal e-values. Moreover, we generalize the notion of rank- and sign-based testing to compact groups, by using a representative inversion kernel. In addition, we characterize e-processes for group invariance for arbitrary filtrations, and provide tools to construct them. We also describe test martingales under a natural filtration, which are simpler to construct. Peeking beyond compact groups, we encounter e-values and e-processes based on ergodic theorems. These nest e-processes based on de Finetti\u2019s theorem for testing exchangeability.<\/jats:p>","DOI":"10.1214\/26-aos2627","type":"journal-article","created":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T08:13:56Z","timestamp":1787040836000},"page":"1843-1869","source":"Crossref","is-referenced-by-count":0,"title":["Measuring evidence against exchangeability and group invariance with E-values"],"prefix":"10.1214","volume":"54","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3784-4480","authenticated-orcid":false,"given":"Nick W.","family":"Koning","sequence":"first","affiliation":[{"name":"Econometric Institute, Erasmus University Rotterdam"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"15","doi-asserted-by":"publisher","unstructured":"KALLENBERG, O.<\/small> (2021). Foundations of Modern Probability<\/i>. Probability Theory and Stochastic Modelling<\/i> 99<\/b>. 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