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It means that for a given $(a_n)^{\\infty}_{n=1}$, $a_n>0$, series $\\sum^{\\infty}_{n=1}a_n\\varphi_n$ is a.e. convergent for each orthonormal sequence $(\\varphi_n)^{\\infty}_{n=1}$ if and only if there exists a measure $m$ on \\[T=\\{0\\}\\cup\\Biggl\\{\\sum^m_{n=1}a_n^2,m\\geq 1\\Biggr\\}\\] such that \\[\\sup_{t\\in T}\\int^{\\sqrt{D(T)}}_0(m(B(t,r^2)))^{-{1}/{2}}\\,dr<\\infty,\\] where $D(T)=\\sup_{s,t\\in T}|s-t|$ and $B(t,r)=\\{s\\in T:|s-t|\\leq r\\}$. The presented approach i"},"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":"1303.4547","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-03-19T10:59:09Z","cross_cats_sorted":[],"title_canon_sha256":"7b6b380b1c56360659a096e825c072de2555baf653e8d21bec5d8a00f67ede26","abstract_canon_sha256":"ff695b25bbd1bb11e38226b8cb00469253eac1d57ea99e86bbef7d16bbd60b59"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:30:26.849855Z","signature_b64":"ZidILSrBcRGBnb1o2OhHDb+6IKlBse3pSFiit6g+g/UBf63F0sUzQ3Ua2CNMF1XiA6zB9UutqW4KT6bqddXPCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"738e6a7b5a82d30772a8cb2dce83ae662918d70f50f59ad74e41269ffb7abb0d","last_reissued_at":"2026-05-18T03:30:26.848999Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:30:26.848999Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The complete characterization of a.s. convergence of orthogonal series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Witold Bednorz","submitted_at":"2013-03-19T10:59:09Z","abstract_excerpt":"In this paper we prove the complete characterization of a.s. convergence of orthogonal series in terms of existence of a majorizing measure. It means that for a given $(a_n)^{\\infty}_{n=1}$, $a_n>0$, series $\\sum^{\\infty}_{n=1}a_n\\varphi_n$ is a.e. convergent for each orthonormal sequence $(\\varphi_n)^{\\infty}_{n=1}$ if and only if there exists a measure $m$ on \\[T=\\{0\\}\\cup\\Biggl\\{\\sum^m_{n=1}a_n^2,m\\geq 1\\Biggr\\}\\] such that \\[\\sup_{t\\in T}\\int^{\\sqrt{D(T)}}_0(m(B(t,r^2)))^{-{1}/{2}}\\,dr<\\infty,\\] where $D(T)=\\sup_{s,t\\in T}|s-t|$ and $B(t,r)=\\{s\\in T:|s-t|\\leq r\\}$. 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