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R be a Lebesgue measurable function. The approximate upper, respectively lower, limit of f at x is defined as aplimsup f (y) := inf{t E R y-x I 0*({z E IRn I f (z) > t}, x) = 0} respectively apliminf f (y) y-ix sup{t E R 10* ({z E IRn I f (z) < t}, x) = 0} . We speak of approximate limit of f at x in case aplim f (y) := aplimsup f (y) = apliminf f (y) y-+x 'Y-+x and we say f to be approximately continuous at x if aplim f (y) = f (x) y-x We then have, compare Sec.

If f : X -* [-oo, +oo] is p-summable, then for every e > 0 there exists b > 0 such that Jfdi < C A for all p-measurable sets A with p(A) < J. Definition 10. We say that fk converge to f in L1(X; p) if and only if f f-fd 0. , Llconvergence implies convergence in measure. e. in a set of finite measure implies convergence in measure. 1. General Measure Theory 8 Proposition 4. Let {fk} be a sequence of p-measurable functions, which converges to f in measure p. Then there exists a subsequence { fk, } such that fk.

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