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Measurable Functions

Suppose u(x) : X--> R
v(x) : X --> R
Both u(x) and v(x) are measurable

Let f(x) : x --> R^2
f(x) = (u(x), v(x) )

Then f (x) is measurable

Now prove a generalization of the above.
That is, prove:

if u_1(x) : X--> R
u_2(x): X--> R
u_n(x) : X--> R
u_1,...,u_n are measurable
and if f(x) : X --> R^n

then prove that f = ( u_1,u_2,...,u_n) is measurable
Please I want a detialed proof and justify every step you make, I need this next hour if possible..thanks.

Solution Summary

Measurable functions are investigated.