The present book is alogical consequence of' ZF - Axiomof Foundation + Axiomof Countable Choice ', is divided into seven chapters and is addressed to students in economics or mathematics who are willing to acquire basic knowledge in industrial and applied mathematics.The first chapter addresses the Frechet derivative and the second chapter addresses the Gel'fand- Pettis integral,which is used to define the stochastic integral,while the third chapter provides basic knowledge on descriptive set theory.The fourth chapter addresses fixed point theory and, in particular, Brouwer's theorem and Kakutani's theorem. The fifth chapter addresses games in strategic form, which model static strategic interactions, and the concept of Nash equilibrium in pure strategies,which is used to make predictions about their outcome.The sixth chapter addresses mixed extensions of game sinstrategic form, which again model static strategic interactions,where the players randomize over their strategies, and the concept of Nash equilibrium in mixed strategies, which is used to make predictions about their outcome.The seventh chapter first gives a brief introduction to Bayesian games,which model static strategic interactions within complete information,and the concept of Bayesian Nash equilibrium,which is used to make predictions about their outcome, and second it gives abrief introduction to extensive games,which model dynamic strategic interactions, and the concepts of Nash equilibrium and subgame perfect Nash equilibrium,which are used to make predictions about their outcome. Inaddition,the exercises complement the main text, as, for example, giving a brief introduction to extensive games with imperfect information, which model dynamic strategic interactions with imperfect information,where the concept of Nash equilibrium in mixed strategiesis used to make predictions about their outcome, and so do the appendices,while for any numbered item in the text the reader is referred to its bibliography. Via two laws, the first on March 8, 1994 and the second on October 13, 1998, the Hellenic Parliament recognizes the Genocide of the Greeks of Pontos, so the present book is dedicated to their memory.