By A N Kolmogorov

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L-measurable. Proof. Sincefg = [(f + g)2 — (f — g)2], the assertion follows from Theorems 5 and 6, and the fact that = t2 is a con- tinuous function. Exercise. Show that if f( x) is measurable and does not take on the value zero, then 1 /f( x) is also measurable. In studying measurable functions one can often neglect the values of a function on a set of measure zero. In connection with this let us introduce the following Two functions f and g, defined on one and the same measurable set E, are called equivalent (denoted by f g), if g(xfl = 0.

0, is 2. SYSTEMS OF SETS 19 2. Systems oF Sets Before proceeding to the general theory of Sets we Shall first give some information concerning systems of sets, which supplements the elements of set theory discussed in Chapter I of Volume I. shall call any set, the elements of which are again certain sets, a system of sets. As a rule we shall consider systems of sets, each of which is a subset of some fixed set X. We shall usually denote systems of sets by Gothic letters. Of basic interest to us will be systems of sets satisfying, with respect to the operations introduced in Chapter I, §1 of Volume I, certain definite conditions of closure.

26 I. MEASURE THEORY Rings whose elements satisfy condition (*) are called "Boolian" rings. Each Boolian ring can be realised as a ring of sets with the operations B and A ii B (Stone). A 3. Measures on Semirings. Continuation oF a Measure From a Semiring to the Minimal Ring over it In §1, when considering measure in the plane, we Started from the measure of a rectangle (area) and then extended the concept of measure to a wider class of sets. The results as well as the methods given in §1 have a quite general character and can be generalized to measures defined on arbitrary sets without essential changes.