Course in Probability Theory

Distribution function1 .1 Monotone functionsWe begin with a discussion of distribution functions as a traditional wayof introducing probability measures . It serves as a convenient bridge fromelementary analysis to probability theory, upon which the beginner may pauseto review his mathematical background and test his mental agility . Some ofthe methods as well as results in this chapter are also useful in the theory ofstochastic processes .In this book we shall follow the fashionable usage of the words "positive","negative", "increasing", "decreasing" in their loose interpretation .For example, "x is positive" means "x > 0" ; the qualifier "strictly" will beadded when "x > 0" is meant . By a "function" we mean in this chapter a realfinite-valued one unless otherwise specified .Let then f be an increasing function defined on the real line (-oc, +oo) .Thus for any two real numbers x1 and x2,(1) x1 < x2 = f (XI) < f (x2) •We begin by reviewing some properties of such a function . The notation"t 'r x" means "t < x, t - x" ; "t ,, x" means "t > x, t -a x" .