Exercises and Solutions Manual for Integration and Probability

'I Measurable Spaces and Integrable Functions.- 1 ?-algebras and partitions.- 2 r-families.- 3 Monotone classes and independence.- 4 Banach limits.- 5 A strange probability measure.- 6 Integration and distribution functions.- 7 Evaluating $$ \sum\nolimits_{n = 1}^\infty {\frac{{{{\left( { - 1} \right)}^n}}}{{{n^2}}}} $$.- 8 Monotone convergence.- 9 Vector integration.- 10 Convergence in measure and composition of functions.- 11 Principle of separation of variables.- 12 The Cauchy-Schwarz inequality.- 13 Test that X ? Y almost everywhere.- 14 Image of a measure.- 15 Primitives of square integrable functions.- II Borel Measures and Radon Measures.- 1 Positive measures on an open interval.- 2 Distribution functions.- 3 Convexity and growth.- 4 Convexity and measure.- 5 Integral representation of positive convex functions (0, ?).- 6 Integral representations of Askey functions.- 7 Gauss's inequality.- 8 Integral of a decreasing function.- 9 Second mean value theorem for integrals.- 10Variance of a distribution on [0, 1].- 11 Variance of the distribution of a convex function on [0, 1].- 12 Rational functions which preserve Lebesgue measure.- 13 A measure on the half-plane.- 14 Weak convergence and moments.- 15 Improper integrals and Lebesgue measure.- 16 $$ \int_0^\infty {\frac{{\sin x}}{x}} dx,\int_0^\infty {\left( {\cos ax - \cos bx} \right)} \frac{{dx}}{x},\int_0^\infty {\left( {\cos ax - \cos bx} \right)} \frac{{dx}}{{{x^2}}} $$.- 17 Comparisons between different Lp spaces.- 18 Differentiation under the integral sign.- 19 Laplace transform of a measure on [0,+?).- 20 Comparison of vague, weak, and narrow convergence.- 21 Weak compactness of measures.- 22 Vague convergence and limit of µn (0).- 23 Vague convergence and restriction to a closed set.- 24 Change of variables in an integral.- 25 Image of a measure and the Jacobian.- III Fourier Analysis.- 1 Characterizations of radial measures.- 2 Radial measures and independence.- 3 Area of the sphere.- 4 Fourier transform