ii let f e r be a measurable function if f is integrable then fis inte
1) Let E be a set. Define a ringon E and a o-algebraon E. (b) Let A be the collection of subsets of R consisting of all finite unionsA = (a1,b1] U...U (an, bn] of disjoint semi-closed intervals. Using yourdefinition in (a) check whether A is a ring and whether A is a o-algebra[5 marks]on R. (c) Let (E, &,µ) be a measure space, and B(R) the Borel o-algebra on R.For each of the following statements determine whether it is true orfalse. Justify your answers with a proof or a counterexample. (i) Let f : E → R be a (E – B(IR)) measurable function. Then f isintegrable. (ii) Let f : E → R be a measurable function. If f is integrable, then |f|is integrable. (iii) Let fn : E → R be a sequence of measurable functions and assumethat flim fn a.e. on E. Then ƒ is measurable. (iv) Let fn : E → R be a sequence of integrable functions and assumethat flim fn a.e. on E. Then f is integrable.
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