(1)Is there a real-valued function f satisfies
the set {(x,f(x)),x belongs to R} is a second category subset of R2?
(2)Is there a real-valued function f satisfies
the set {(x,f(x)),x belongs to R} is non-measurable in the Lebesgue sense?
(1)Is there a real-valued function f satisfies
the set {(x,f(x)),x belongs to R} is a second category subset of R2?
(2)Is there a real-valued function f satisfies
the set {(x,f(x)),x belongs to R} is non-measurable in the Lebesgue sense?
Hint: For any $\displaystyle A\subset \mathbb{R}$ non Lebesgue measurable, choose the indicator function $\displaystyle \chi_A:\mathbb{R}\to \mathbb{R}$
$\displaystyle \chi_A (x)=\begin{Bmatrix} 1 & \mbox{ if }& x\in A\\0 & \mbox{if}& x\not \in A\end{matrix}$
Hint: Use the Baire's Category Theorem: Every complete metric space is of second category.Is there a real-valued function f satisfies the set {(x,f(x)),x belongs to R} is a second category subset of R2?