Which parts of measure theory should a student of mathematics (statistics, finance, etc.) know in order not to be frustrated by an advanced and rigorous course on probability theory? An obvious answer is that the standards on abstract integration and construction of a measure are not enough when nowadays, even at the undergraduate level, topics such as martingales and Brownian motion are often included. The text is written in an economic way, but space is left also for heuristics and history with the aim to introduce the reader to probabilistic inventiveness and thinking. Integration is developed first via extended linear functionals on spaces of measurable functions (in Appendix A); then follow basic probabilistic topics like independence, conditioning, martingales and Fourier transforms. The central limit theorems, convergence to Brownian motion, strong representations, couplings and the law of iterated logarithm receive a deep and detailed treatment to provide the reader with a solid information on more recent developments in the Gaussian branch of probability. The text includes also sections that may be considered really advanced (the martingale characterization of Brownian motion, option pricing via equivalent martingale measures and the disintegration procedures being a sample of them). The formal mathematical material is accompanied by solved exercises and by a collection of frequently challenging problems. A really useful book for everybody who faces students interested in modern probability without a preliminary knowledge of measure and integration theory.