07-03. Measure theory and probability - Construction of the Lebesgue integral.

This video gives a four-step construction of the Lebesgue integral and more generally the integral of a measurable function with respect to a positive measure for (1) indicator functions, (2) simple functions, (3) positive functions and finally (4) general measurable functions. From the point of view of the Lebesgue integral, the probability that a number chosen uniformly at random in the unit interval is rational exists and we prove that this probability is equal to zero. This is Section 1.2 of my Stochastic Modeling book.

[myroslavkryven]

Is it obvious that the Lebesgue integral of the characteristic function of a set over this set is the measure of the set? I refer to 4:56 in the video.

[Juanitoto]

in 4.) can't just EITHER X(+) or X(-) be bounded? Then the integral is either (inf - a) or (b - inf).

[qiaohuizhou]

Hi, I am wondering shouldn't the density function be 1_[0, 1] instead of 1_Q? A density function of uniform 0, 1 distribution is 1 on [0, 1], not 1 on Q and 0 on Q^c... though the outcome won't change