| 31. | In practice, some authors use "'measurable functions "'to refer only to real-valued measurable functions with respect to the Borel algebra.
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| 32. | Then there is a measure space and a real-valued essentially bounded measurable function on and a unitary operator such that
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| 33. | Similarly P _ nf converges to \ mathbb { E } f almost surely for a fixed measurable function f.
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| 34. | Recall that it follows from Lusin's theorem that a Lebesgue-measurable function is approximately continuous almost everywhere ( and conversely ).
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| 35. | We have defined the integral of " f " for any non-negative extended real-valued measurable function on " E ".
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| 36. | In practice, some authors use "'measurable functions "'to refer only to real-valued measurable functions with respect to the Borel algebra.
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| 37. | A function between two measurable spaces is called a measurable function if the preimage of every measurable set is measurable.
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| 38. | I know for a bounded measurable function, they define the Lebesgue integral but never what integrable means specifically for such functions.
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| 39. | In particular, the above result implies that is included in, the sumset of and in the space of all measurable functions.
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| 40. | Of course, the zero " norm " is "'not "'truly a norm, because it is not Lebesgue space of measurable functions.
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