Week 4 Coursework Test

Let $p, q > 1$ satisfy $\frac{1}{p} + \frac{1}{q} = 1$ and fix a measure space $(X, B, μ)$ . Fix $f$ in $L^{p} (X, B, μ)$ and define $ξ_{f} : L^{q} (X, B, μ) \to R$ by $ξ_{f} (g) = \int f \cdot g d μ$ for all $g \in L^{q} (X, B, μ)$ . Verify that $ξ_{f}$ is well-defined and linear.
Let $μ$ be Lebesgue measure on $R$ .
1. If $f \in L^{1} (R, Bor (R), μ)$ must $g (x) = x f (x)$ belong to $L^{1} (R, Bor (R), μ)$ ?
2. If $f : R \to R$ is differentiable must $f^{'}$ belong to $L^{1} (R, Bor (R), μ)$ ?