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