I am currently trying to learn about modes of convergence in Measure theory and am struggling to understand the difference between definitions of uniform and pointwise.
I have either proofs or counter-examples for the following in my notes:
For $f, f_n \in L^p (µ)$ for $p< \infty$, $f_n(x) \to f(x)$ uniformly a.e.(µ) implies $f_n \to f$ in $L^p (µ)$.
For $f, f_n \in L^p (µ)$ for $p< \infty$, $f_n → f$ in $L^p (µ)$ implies $f_n(x) → f(x)$ uniformly a.e.(µ)
Let $f, f_n ∈ L^p(µ)$ where $µ(X) < ∞$. If $f_n → f$ uniformly a.e. then $f_n → f$ in $L^p(µ)$.
Let $f, f_n ∈ L^p(µ)$ with $f_n → f$ in norm. Then there is a subsequence $(f_{n_k})_k$ of $(f_n)$ such that $f_{n_l}(x) → f(x)$ a.e.(µ).
I am struggling to figure out which, if any, of these match up with these past paper exam qns:
$(fn)_n$ is a sequence of functions in $L^p(µ)$. Give a proof of, or a counter-example to, each of the following statements.
(i) $f_n → 0$ uniformly implies $f_n → 0$ pointwise.
(ii) $f_n → 0$ uniformly implies $f_n → 0$ in $L^p(µ)$ (that is, $\Vert f_n \Vert_p → 0$).
(iii) $f_n → 0$ uniformly on a finite measure space implies $f_n → 0$ in $L^p(µ)$ (that is, $\Vert f_n \Vert_p → 0$).
(iv) $f_n → 0$ in $L^p(µ)$ implies $f^n → 0$ pointwise.
Any help would be appreciated.