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It remains to extract out the pure point components. This we do by a similar procedure as above. Indeed, by arguing as before we may assume (after passing to a subsequence as necessary) that the quantities µn {x : µn ({x}) ≥ 1/j} converge to a limit dj for each positive integer j, that the dj themselves converge to a limit d, and that there exists a sequence jn → ∞ such that µn {x : µn ({x}) ≥ 1/jn } converges to d. 2. Signed measures 25 1/jn } and {x : µn ({x}) ≥ 1/jn } respectively, one can verify the remaining claims by arguments similar to those already given.

It remains to extract out the pure point components. This we do by a similar procedure as above. Indeed, by arguing as before we may assume (after passing to a subsequence as necessary) that the quantities µn {x : µn ({x}) ≥ 1/j} converge to a limit dj for each positive integer j, that the dj themselves converge to a limit d, and that there exists a sequence jn → ∞ such that µn {x : µn ({x}) ≥ 1/jn } converges to d. 2. Signed measures 25 1/jn } and {x : µn ({x}) ≥ 1/jn } respectively, one can verify the remaining claims by arguments similar to those already given.

Depending on which of the three structures X, X , µ of the measure space one wishes to emphasise, the space Lp (X, X , µ) is often abbreviated Lp (X), Lp (X ), Lp (X, µ), or even just Lp . Since 6One could also take a more abstract view, dispensing with the set X altogether and defining the Lebesgue space Lp (X , µ) on abstract measure spaces (X , µ), but we will not do so here. Another way to think about elements of Lp is that they are functions which are “unreliable” on an unknown set of measure zero, but remain “reliable” almost everywhere.

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