can any help me how to proceed with induction to solve this problem.
let I subset from R be an interval of the form [a, b],[a, b),(a, b] or (a, b) for a, b belong to R satisfying a less then or equals b . Define the length of such an interval I to be
b-a . A box in R^d is a Cartesian product B=I_1 x ...x I_d if intervals in R . Define the length of such a box B to be B=|I_1|x...|I_d| . An elementary set in R^d is a set which is the union of a finite number of boxes.
(a) Prove that if E, F subsets from R^d are elementary sets , then E union F and E intersection F are also elementary sets.
(b) Prove that every elementary set in R^d can be written as a finite union of disjoint boxes.
(c) Prove that for an interval I as above ,
|I| = lim_n goes to infinity 1/n# (I intersection 1/n Z),
where 1/n Z ={k/n , k belongs to Z} and #A denotes the cardinality of a finite set A .
(d) Prove that for a box B subset from R^d , |B|= lim_n goes to infinity 1/n #(B intersection 1/n Z^d ).
(e) for any elementary set E subset from R ^d , define m(E)=|B_1| +...+|B_n| , where E= B_1 union ...union B_n is any decomposition of E into disjoint boxes B_1,...,B_n subset from R^d . Prove that m(E) = lim_ n goes to infinity #(E intersection 1\n Z ) , hence the value of m(E) does not depend on the particular decomposition chosen .
let I subset from R be an interval of the form [a, b],[a, b),(a, b] or (a, b) for a, b belong to R satisfying a less then or equals b . Define the length of such an interval I to be
b-a . A box in R^d is a Cartesian product B=I_1 x ...x I_d if intervals in R . Define the length of such a box B to be B=|I_1|x...|I_d| . An elementary set in R^d is a set which is the union of a finite number of boxes.
(a) Prove that if E, F subsets from R^d are elementary sets , then E union F and E intersection F are also elementary sets.
(b) Prove that every elementary set in R^d can be written as a finite union of disjoint boxes.
(c) Prove that for an interval I as above ,
|I| = lim_n goes to infinity 1/n# (I intersection 1/n Z),
where 1/n Z ={k/n , k belongs to Z} and #A denotes the cardinality of a finite set A .
(d) Prove that for a box B subset from R^d , |B|= lim_n goes to infinity 1/n #(B intersection 1/n Z^d ).
(e) for any elementary set E subset from R ^d , define m(E)=|B_1| +...+|B_n| , where E= B_1 union ...union B_n is any decomposition of E into disjoint boxes B_1,...,B_n subset from R^d . Prove that m(E) = lim_ n goes to infinity #(E intersection 1\n Z ) , hence the value of m(E) does not depend on the particular decomposition chosen .