prove sequences - 2 problems

(See attached file for full problem description) --- 1) a. Let with . If is a measurable subset of R, prove that and are measurable. b. let E =[0,1]Q. Prove that E is measurable and (E)=1. c. let P denote the cantor set in [0,1]. Prove that 2) If E R, prove that ther exists a sequence { } of open sets with E for all n such that = . --- (See attached file for full problem description)

Measurable functions

1) a. If g is measurable on [a, b] with g(x) 0 for all x [a, b], prove that 1/g is measurable on [a, b]. b. prove that every continuous real- valued function f on [a, b] is measurable. c. if f is differentiable on [a, b], prove that f' is measurable on [a, b]

see attachement

(See attached file for full problem description) --- 1) a. Let f be defined on [a, b] by f(x) prove directly that f is measurable b. let E be measurable subset of R, and let f be measurable function on E Define the function f and f on E as follows: f (x) = max { f(x), 0}, and f (x) = max{ -f(x), 0}, 1b. prove directly that f and f are nonnegative measurable function on E with f(x) = f (x) - f (x). 2b. prove that = and that is measurable. 3b. if f is areal- valued function on [a, b] such that is measurable on [a, b] is f measurable on [a,b]? 4b. if f(x)= + cos(x), x [0, 2 ], find f (x) and f (x). --- (See attached file for full problem description)

Statistics

The average numbers of home runs hit by the Boston Red Sox per game are: 2 divided by 3 = .66 5 divided by 2 = 2.5 6 divided by 1 = 6 7 divided by 0 = 0 Is that correct using the weighted approach?

A thin glass pipe with internal diameter of 3mm was probed into the heart tissue (membrane) and air pressure was applied

A thin glass pipe with internal diameter of 3mm was probed into the heart tissue (membrane) and air pressure was applied... See attached file for full problem description.