Nonlinear expansion and development of Andre Andrey Kolmogorov's probability theory and mathematical expectation theory. The nonlinear mathematical expectation theory is the nonlinear extension and development of Andre Andrey Kolmogorov's probability theory and mathematical expectation theory. It is one of the frontier research topics and research hotspots in the fields of probability and stochastic analysis, financial mathematics and so on, and has a wide application prospect in the fields of economy and finance.