function named “g_c_d” that takes two positive integer arguments…

Question Answered step-by-step function named “g_c_d” that takes two positive integer arguments…  function named “g_c_d” that takes two positive integer arguments and returns as its value the greatest common divisor of those two integers. If the function is passed an argument that is not positive (i.e., greater than zero), then the function should return the value 0 as a sentinel value to indicate that an errorSuppose that µ and µ ? are translation invariant measures on the Borel ??algebra of Rd both assigning the same (finite) measure to the unit box [0, 1]d . We will show that then µ = µ ? . Let L denote the set of all Borel sets A ? R n for which µ(A) = µ ? (A). By hypothesis, [0, 1]d ? L. It seems reasonable to conclude from this that the set P of all boxes [a1, b1] × · · · × [ad,bd], with rational ai and bi , would belong to L. Let us accept this for now; i.e. suppose P ? L. Now if we can show from this that L contains the ??algebra generated by P then we would be done, because the ??algebra generated by ? is the Borel ??algebra. (This follows from two observations : (i) each box in P is the intersection of open sets : [a1, b1] × · · · × [ad,bd] = k?1 µ a1 ? 1 k ,b1 + 1 k ¶ × · · · × µ ad ? 1 k ,bd + 1 k ¶ and (ii) every open U subset of Rd is the union of small boxes [a1,b1] × · · · × [ad,bd] with rational endpoints and centered at the rational points in U.) Thus L would in fact be the whole Borel ??algebra. That is, µ(A) = µ ? (A) for every Borel set A. Thus the key tool would be the result that L contains the ??algebra generated by P. This will, essentially, be proved by the ? ? ? theorem. There are some technical problems involved which will be settled later. 6.0. Definition. Let P and L be collections of subsets of a set X. The collection P is called a ??system if it is closed under finite intersections; i.e. if A,B ? P then A ? B ? P: P is a ?-system if A ? B ? P for all A,B ? P The collection L is called a ??system if the following hold : (L1) ? ? L; (L2) if A ? L then A c ? L; (L3) L is closed under countable disjoint unions; i.e. if A1,A2,… ? L and if Ai?Aj = ? for every i 6= j, then ? ? j=1Aj ? L. 6.1. Dynkin’s ? ? ? Theorem. Let P be a ?-system of subsets of X,and L a ?-system of subsets of X. Suppose also that P ? L. Then : ?(P) ? L, i.e. L contains the ?-algebra ?(P) generated by P. We will do the proof later but let us apply it to prove the uniqueness of LebesgueIn the code fragment below, the programmer has almost certainly made an error in the first line of the conditional statement. a. What is the output of this code fragment as it is written? b. How can it be corrected to do what is the programmer surely intended?  Engineering & Technology Computer Science SCIENCE 101 Share QuestionEmailCopy link Comments (0)