1.) Use the definition of the outer measure to show that m*((0,2)) = 2 2.) Let...
Please prove Problem 11 & 12 carefully (note that m represents Lebesgue measure & m* represents Lebesgue outer measure): 11. Let E c Rn be an arbitrary subset. Show that for all є > 0 there exists an open set G containing E with m(G) m"(E) +e. 12. Let E C Rn be a measurable subset. Show that for all € > 0 there exists an open set G containing Ewith m (G\ E) < є. 11. Let E c...
se on a wcm*** *u 2. [2 points] Let )* be an outer measure on 2%. Show that a set A € 2X is 1*-measurable if and only if VU CA and V CA 1*(U UV) = 1*(U) + 1*(V). 3. [2 points) Let u* be the outer measure generated by a measure u defined on a ring R, and let S denotes the set of all p*-measurable sets. Show that or(R) CO(R) CS.
3. Let E E Lm* (Lebesgue measurable set). Prove that there exist a set G (a countable intersection of open sets), and a set F (a countable union of closed sets) such that F CE C G and m* (F) the Lebesgue measure of a set Hint: The Lebesgue measure can be calculated in terms of open and closed sets m* (E) m* (G), where m* denotes 3. Let E E Lm* (Lebesgue measurable set). Prove that there exist a...
3. (3 pts) Let V (0,1be the Vitali set introduced in the lectures. Let u be the outer measure on Rwhose restriction to the u-measurable sets gives rise to the Lebesgue measure. Then choose the correct answer): (a) H(V) is not defined; (b) H(V) = 0; (c) H(V) >0. Justify your answer (use notations and results from the lectures): .
equivalent 4. Let E C R. Prove that the following statements are (a) E is Lebesgue measurable (b) Given e> 0, there exist m* denotes the Lebesgue measure of a set (c) Given e 0, there exist a closed set F such that F C E and m* (E- F) < E. (d) There exists a set G (a countable intersection of open sets) such that E C G and m* (G - E) 0 (e) There exists a set...
equivalent 4. Let E C R. Prove that the following statements are (a) E is Lebesgue measurable (b) Given e> 0, there exist m* denotes the Lebesgue measure of a set (c) Given e 0, there exist a closed set F such that F C E and m* (E- F) < E. (d) There exists a set G (a countable intersection of open sets) such that E C G and m* (G - E) 0 (e) There exists a set...
equivalent 4. Let E C R. Prove that the following statements are (a) E is Lebesgue measurable (b) Given e> 0, there exist m* denotes the Lebesgue measure of a set (c) Given e 0, there exist a closed set F such that F C E and m* (E- F) < E. (d) There exists a set G (a countable intersection of open sets) such that E C G and m* (G - E) 0 (e) There exists a set...
1) Show that if U is a non-empty open subset of the real numbers then m(U) > O. 2) Give an example of an unbounded open set with finite measure. Justify your answer, 3) If a is a single point on the number line show that m ( a ) = O. 4) Prove that if K is compact and U is open with K U then m(K) m(U). 5) show that the Cantor set C is compact and m(C)...
(11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function which is integrable on a set A E A. Let f, : O -> R* be a sequence of measurable functions such that g(x) < fn(x) < fn+1(x), for all E A and n E N. Prove that lim fn d lim fn du noo A (11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function...
please explain the steps you take 2. Let M be the set of all measurable sets in R, and let d be our semi-metric, show that (M, d) is complete: If (An)1 is a Cauchy sequence (with our semi- metric d) then there is a measurable set A EM such that lim, too d(An, A) 0. 2. Let M be the set of all measurable sets in R, and let d be our semi-metric, show that (M, d) is complete:...