# B x b 1 m b b f y f x dy 0 math 212a fall 2017 yum

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(B)0xB1m(B)B|f(y)f(x)|dy= 0.
Math 212a (Fall 2017) Yum-Tong Siu10Givenε >0, there existsrQrsuch that|f(x)r|< ε. From the triangularinequality1m(B)B|f(y)f(x)|dy1m(B)B|f(y)r|dy+|rf(x)|it follows thatlim supm(B)0xB1m(B)B|f(y)f(x)|dylimm(B)0xB1m(B)B|f(y)r|dy+|rf(x)|= 2|rf(x)|<2ε,which implieslimm(B)0xB1m(B)B|f(y)f(x)|dy= 0due to the arbitrariness ofε.Points of Lebesgue Density of Measurable Set.For a measurable setEinRd,a pointxofRdis said to be apoint of Lebesgue densityofEif and only iflimm(B)0xBm(BE)m(B)= 1(whereBis a closed ball). Equivalently, a pointxofRdis a point of Lebesguedensity ofEif and only if for everyα <1 close to 1 and every ball ofsuﬃciently small radius containingx,m(BE)α m(B).When we apply tof=χEthe statement that almost every point belongsto the Lebesgue set of a locally integrable function, we conclude that almostevery point ofEis a point of Lebesgue density ofEand almost every pointof the complement ofEis not a point of density ofE.We now discuss the problem of pointwise convergence of the Fourier seriesof a locally integrable functionfonRwith period 2πin which the Lebesgueset of an integrable function plays a rule. When we talk about the conver-gence of a series, what is meant is the convergence of a sequence constructedfrom the series, which is the sequence of partial sums of the series. There arealso other ways of constructing a sequence from the given series, for example,the Ces`aro sums and the Abel sums which we now explain. We look at thisquestion of the construction of sequences from a given series from a moregeneral viewpoint.
Math 212a (Fall 2017) Yum-Tong Siu11Convergence and Boundedness of Series with Different Sequences of Weights.For a seriesnan, the question of convergence or boundedness deals witha sequence constructed from the series, for example, the sequence of partialsumssn=nk=0an. When the indexnof the seriesnangoes from−∞to, the partial sumsnmay mean thesymmetricorprincipalpartial sum,which means thatsn=nk=nan. Instead of the partial sumsn=nk=0an,the Ces`aro sumσn=1nn1k=0sk=n1k=0nknakis used as the sequence. Abel considers also a function of 0< r <1 (insteadof a sequence which a function ofN) which isA(r) =nr|n|an.The convergence of the sequence of partial sumssnis known as the conver-gence of the seriesnan. The convergence of the sequence of Ces`aro partialsumsσnis known as the convergence of the seriesnanin the sense ofCes`aro. The convergence ofA(r) asr1is known as convergence of theseriesnanin the sense of Abel.