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By P Cassou-Nogues, Taylor

ISBN-10: 0817633502

ISBN-13: 9780817633509

Книга Elliptic capabilities and earrings of Integers Elliptic services and jewelry of Integers Книги Математика Автор: Ph Cassou-Nogues Год издания: 1987 Формат: djvu Издат.:Birkhauser Страниц: 198 Размер: 3,1 ISBN: 0817633502 Язык: Английский0 (голосов: zero) Оценка:Elliptic services and earrings of Integers

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Example text

For a Borel set A we will use the notations H (A) and H p p s(A). Measure on the Real Line 50 Definition 15: The Hausdorff dimension of Ε is inf[p: Hp*(E) = 0 ] . By Corollary 4 to Theorem 22 Hausdorff dimension is a well defined number in the interval [0,1] for any set A in the real line. Also, except for the trivial case of sets Ε with H$(E) = 0 for all p , we have that Hausdorff dimension equals sup[p: H^(E) = ° o ] . Theorem 27: The Hausdorff dimension of the Cantor-like setP^ is — log 2/log £.

E. e. Then g is measur­ able. Proof: [x: f(x) > α] Δ [x: g(x) > a] ç [x: f{x) # * ( * ) ] and the result follows immediately from Example 4, p. 30, and Example 5, p. 31. e. to / ; then/is measurable, since/ = lim s u p a . e . t Example 14: If / is a measurable function, then so are f r = -min(/,0). = max if, 0 ) and Solution: Example 10 and Theorem 14, ( i ) and ( i i ) , give the result. Example 15: The set of points on which a sequence of measurable functions { / „ } converges, is measurable.

Proof: From Definition 2, R E M and the symmetry in Definition 2 between Ε and CE implies that if E E M then CEE Μ· So it remains to be shown that if oo { £ ) } is a sequence of sets in M then Ε - \ J Ε/Ε M. Let A be an arbitrary set. 5) again (with Ε replaced by E and A by Α Π ΟΕΊ ) we have 2 m*(A) = m*(A Π F, ) + m*(A ΠΕ Π0Ε ) 2 1 + m*(A n CE", η CE ). 2 Sec. 2] Measurable Sets 31 Continuing in this way we obtain, for η > 2, m*(Â) = m*(A η F , ) + X m*(A Π Ε O f ) C £ » + m*(A η Π C Ê » { ί=2 / < / /=1 η η = m*{A O F , ) + Χ m%4 Π £ , Π c l ^ F / ) + m*(4 Π C U £"/) /<ί 1=2 /=ι > m%4 O F , ) + Χ m*(A η Et Π C^JEA + m*(A η C Ù ΕΛ, i=2 /

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Elliptic Functions and Rings of Integers by P Cassou-Nogues, Taylor


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