{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T11:54:34Z","timestamp":1759146874904},"reference-count":30,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":101,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2013,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We give a full description of the structure under inclusion of all finite level Borel classes of functions, and provide an elementary proof of the well-known fact that not every Borel function can be written as a countable union of \u03a3<jats:sub arrange=\"stack\"><jats:italic>\u03b1<\/jats:italic><\/jats:sub><jats:sup arrange=\"stack\">0<\/jats:sup>-measurable functions (for every fixed 1 \u2264 <jats:italic>\u03b1 &lt; \u03c9<\/jats:italic><jats:sub>1<\/jats:sub>). Moreover, we present some results concerning those Borel functions which are <jats:italic>\u03c9<\/jats:italic>-decomposable into continuous functions (also called countably continuous functions in the literature): such results should be viewed as a contribution towards the goal of generalizing a remarkable theorem of Jayne and Rogers to all finite levels, and in fact they allow us to prove some restricted forms of such generalizations. We also analyze finite level Borel functions in terms of composition of simpler functions, and we finally present an application to Banach space theory.<\/jats:p>","DOI":"10.2178\/jsl.7804150","type":"journal-article","created":{"date-parts":[[2014,1,5]],"date-time":"2014-01-05T20:33:25Z","timestamp":1388954005000},"page":"1257-1287","source":"Crossref","is-referenced-by-count":8,"title":["On the Structure of Finite Level and <i>\u03c9<\/i>-Decomposable Borel Functions"],"prefix":"10.1017","volume":"78","author":[{"given":"Luca Motto","family":"Ros","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200126556_ref030","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1007\/BF01876046","article-title":"Baire 1 functions which are not countable unions of continuous functions","volume":"66","author":"van Mill","year":"1995","journal-title":"Acta Mathematica Hungarica"},{"key":"S0022481200126556_ref027","unstructured":"Semmes B. , A game for the Borel functions, Ph.D. thesis, ILLC, University of Amsterdam, Amsterdam, Holland, 2009."},{"key":"S0022481200126556_ref025","doi-asserted-by":"crossref","first-page":"195","DOI":"10.14321\/realanalexch.35.1.0195","article-title":"A new proof of a theorem of Jayne and Rogers","volume":"35","author":"Ros","year":"2010","journal-title":"Real Analysis Exchange"},{"key":"S0022481200126556_ref024","volume-title":"Mathematical Structures in Computer Science","author":"Ros"},{"key":"S0022481200126556_ref023","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1002\/malq.200910124","article-title":"Game representations of classes of piecewise definable functions","volume":"57","author":"Ros","year":"2011","journal-title":"Mathematical Logic Quarterly"},{"key":"S0022481200126556_ref021","doi-asserted-by":"crossref","first-page":"29","DOI":"10.14321\/realanalexch.34.1.0029","article-title":"A new characterization of Baire class 1 functions","volume":"34","author":"Ros","year":"2009","journal-title":"Real Analysis Exchange"},{"key":"S0022481200126556_ref020","first-page":"27","volume":"74","author":"Ros","year":"2009","journal-title":"Borel-amenable reducibilities for sets of reals"},{"key":"S0022481200126556_ref019","volume-title":"Topology","volume":"I","author":"Kuratowski","year":"1966"},{"key":"S0022481200126556_ref018","doi-asserted-by":"crossref","first-page":"206","DOI":"10.4064\/fm-22-1-206-220","article-title":"Sur une g\u00e9n\u00e9ralisation de la notion d'hom\u00e9omorphie","volume":"22","author":"Kuratowski","year":"1934","journal-title":"Fundamenta Mathematicae"},{"key":"S0022481200126556_ref017","first-page":"192","article-title":"Sur les fonctions premi\u00e8res measurables B","volume":"4","author":"Keldys\u0306","year":"1934","journal-title":"Soviet Mathematics Doklady"},{"key":"S0022481200126556_ref015","doi-asserted-by":"publisher","DOI":"10.14321\/realanalexch.38.1.0121"},{"key":"S0022481200126556_ref014","first-page":"177","article-title":"First level Borel functions and isomorphisms","volume":"61","author":"Jayne","year":"1982","journal-title":"Journal de Math\u00e9matiques Pures et Appliqu\u00e9es. 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