Thread: Existence of Real-Valued Functions Satisfying Certain Properties.

1. Existence of Real-Valued Functions Satisfying Certain Properties.

(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?

2. Re: a question about real-valued functions on R

Originally Posted by mathabc
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 $A\subset \mathbb{R}$ non Lebesgue measurable, choose the indicator function $\chi_A:\mathbb{R}\to \mathbb{R}$

$\chi_A (x)=\begin{Bmatrix} 1 & \mbox{ if }& x\in A\\0 & \mbox{if}& x\not \in A\end{matrix}$

Is there a real-valued function f satisfies the set {(x,f(x)),x belongs to R} is a second category subset of R2?
Hint: Use the Baire's Category Theorem: Every complete metric space is of second category.

3. Re: a question about real-valued functions on R

... and $\Gamma (I)=\Delta=\{(x,x):x\in\mathbb{R}\}$ is complete.

And with this, we end the lively conversation.

4. Re: a question about real-valued functions on R

Originally Posted by FernandoRevilla
Hint: For any $A\subset \mathbb{R}$ non Lebesgue measurable, choose the indicator function $\chi_A:\mathbb{R}\to \mathbb{R}$

$\chi_A (x)=\begin{Bmatrix} 1 & \mbox{ if }& x\in A\\0 & \mbox{if}& x\not \in A\end{matrix}$
Just an observation: The graph of that function is a subset of a set of measure 0 (the line y=1).

5. Re: a question about real-valued functions on R

Originally Posted by Jose27
Just an observation: The graph of that function is a subset of a set of measure 0 (the line y=1).
Right, which implies $\mu^* (\Gamma (1_A))=0$ if you consider the complection $\mu^*$ of $\mu$ .

P.S. I suppose you meant a subset of the union of the lines $y=0$ and $y=1$ .