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We say that a function $f\\in L^2(X,\\mu)$ is $\\mu$-mean equicontinuous if for any $\\epsilon>0$ there is $k\\in \\mathbb{N}$ and measurable sets ${A_1,A_2,\\cdots,A_k}$ with $\\mu\\left(\\bigcup\\limits_{i=1}^k A_i\\right)>1-\\epsilon$ such that whenever $x,y\\in A_i$ for some $1\\leq i\\leq k$, one has \\[ \\limsup_{n\\to\\infty}\\frac{1}{n}\\sum_{j=0}^{n-1}|f(T^jx)-f(T^jy)|<\\epsilon. \\] Measure complexity with respect to $f$ is also introduced. It is shown that $f$ is an almost periodic function if and only if $f$ is $\\mu$-mean equicontinuous if and on"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1807.05868","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-07-13T03:00:02Z","cross_cats_sorted":[],"title_canon_sha256":"6fdb78b517704d8ed417c2dddf04116695421eaa05ac39cb65a3dec8bbe610b5","abstract_canon_sha256":"8994ead1fee8a34b68f78687fa5c9915a147d911498bc757c5c033550e2fbf02"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:42.351436Z","signature_b64":"9r9n6DjN7Jt6/MRifjyLfehsQqJ8gGKrlnrfNzQxKJT3AFzDJYTEdaDjU7YtSSvt+dRnqK0lg7mDq1U24AYNAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0aaa982b271d6a4d5c45fd39d282548b540b80bc179a3386687f32a12bcc0587","last_reissued_at":"2026-05-18T00:10:42.350773Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:42.350773Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Measure-theoretic mean equicontinuity and bounded complexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Tao Yu","submitted_at":"2018-07-13T03:00:02Z","abstract_excerpt":"Let $(X,\\mathcal{B},\\mu,T)$ be a measure preserving system. We say that a function $f\\in L^2(X,\\mu)$ is $\\mu$-mean equicontinuous if for any $\\epsilon>0$ there is $k\\in \\mathbb{N}$ and measurable sets ${A_1,A_2,\\cdots,A_k}$ with $\\mu\\left(\\bigcup\\limits_{i=1}^k A_i\\right)>1-\\epsilon$ such that whenever $x,y\\in A_i$ for some $1\\leq i\\leq k$, one has \\[ \\limsup_{n\\to\\infty}\\frac{1}{n}\\sum_{j=0}^{n-1}|f(T^jx)-f(T^jy)|<\\epsilon. \\] Measure complexity with respect to $f$ is also introduced. 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