Explanation of the condition - not independent of the Jacobians of functions.

Real Analysis Jacobians (II) Explanation of the condition - not independent of the Jacobians of functions.

Explanation of the condition - not independent of the Jacobians of functions.

Real Analysis Jacobians (II) Explanation of the condition - not independent of the Jacobians of functions.

real analysis

If f is one-to-one, f, f^-1 are continuous, then f is called a homeomorphism. Now I want you to prove the following: Let f : X -> Y, ( X and Y are topological spaces)be homeomorphism, prove that it establishes one-to-one correspondence between Borel sets in X and Y.

Real Analysis, sup, inf, measurable functions.

------------------------------------------------------------------------------------------- 1). If g_n = Sup f_n, then prove that ( g_n)^-1 ( ( alpha, infinity] ) = union ( n = 1 to infinity) (f_n)^-1((alpha,infinity]). ------------------------------------------------------------------------------------------- 2). Prove that y(x) = inf f_n(x) is a measurable function if all f_n(x) are measurable. ------------------------------------------------------------------------------------------- Please I want very detailed proofs, justify every step and prove every claim you make. Thanks :)

Real Analysis lim sup and lim inf

Let {a_n} and {b_n} be sequences in [-infinity,+infinity] and prove the following assertions: 1). a).Lim sup (as n -> infinity) ( a_n + b_n) less than or equal to lim sup a_n + lim sup b_n ( as n foes to infinity). b).Show by an example that strict inequality can hold. Provided none of the sums is of the form infinity negative infinity 2). If a_n less thaan or equal b_n for all n, then lim inf a_n less than or equal lim inf b_n as n goes to infinity Please I want the answer in the next hour or 2 at most.