Changes in the lecture notes

DateChange
2025-10-28Fix typos from 2025-10-28 and add explanation why the two Radon measures in the proof of Theorem 3.2.3 have the same measurable sets.
2025-10-26Correct wrong inclusion in sheet 7, exercise 3d to L^p ⊂ L^{p,∞}.
2025-10-26Add convolution and approximation of identity section.
2025-10-21Move some exercises from sheet 6 to sheet 7.
2025-10-21Fix mistakes in exercise sheet 6.
2025-10-17Add missing assumption to exercise sheet 6, question 2.
2025-10-17Add section on signed measures.
2025-10-16Proof of Fubini: Properly prove near-measure properties of ρ including the required measurability properties, but do it only for P_1 and P_2 since this is all we use.
2025-10-16In proof of Fubini proof near-measure properties of ρ.
2025-10-12Write chapter on differentiation and integration on ℝ.
2025-10-10Fix wrong question in exercise sheet 4 and write solution.
2025-10-10Finish proof of Fubini.
2025-10-08Define 0*∞ = 0 for integrals of simple functions. Prove that for f >= 0 we have ∫f = 0 if and only if f=0 a.e..
2025-10-07Fix assumptions in integral of simple functions to include simple functions with changing sign. Correct typos noted on 2025-10-07.
2025-10-07Clarify that L_* is Lebesgue measure in Sheet 4 Question 1.
2025-10-07Start proving Fubini.
2025-10-07Finish Lp spaces section.
2025-10-04Move triangle inequality from integral of simple functions to integral of Lebesgue measurable functions.
2025-10-03Do not define integral for simple functions with varying sign and add more details to proofs of basic properties of the integral of nonnegative and of general functions.
2025-10-03In approximation of measurable functions by step functions, insert argument how to go from characteristic functions to simple functions.
2025-09-30Write simpler proof of approximation by simple functions.
2025-09-30Fix definition of set where functions are far apart.