Prove that a constant function is measurable SolutionLet X b

Prove that a constant function is measurable.

Solution

Let X be any set and M={empty set, X}

If f(x) = c for all x element of X, then f-inverse(V)=X if c is an element of V or empty set if c is not an element of V. Since M={empty set, X}, f(x)=c is measurable.

or

another approach:

Let f=C a constant, and to show f is measurable. we can take (a,), we show f1((a,))) is measurable.

But f1((a,)) is just the real line R, and therefore measurable.

That\'s one of the two parts. if ac, then f1[(a,)]=.

since is also measurable, f is measurable.