Do weak convergence or strong convergence in every $L^p$ imply almost-everywhere convergence?

Question

Let $f_n,f\in C(\mathbb R^d)\cap L^1(\mathbb R^d)$ be positive functions such that $$ \int_\mathbb{R^d} f_n(x)\,dx \,=\, \int_\mathbb{R^d} f(x)\,dx$$ and either:

1. for every $\phi\in C_c(\mathbb R^d)$

$$ \lim_{n\to\infty}\int_{\mathbb R^d} \phi(x)\,f_n(x)\,dx \,=\, \lim_{n\to\infty}\int_{\mathbb R^d} \phi(x)\,f(x)\,dx\;;$$

or:

2. for every $K$ compact subset of $\mathbb R^d$, every $p\geq1$ $$ \lim_{n\to\infty} \int_K |f_n(x)-f(x)|^p\,d x \,=\, 0\;.$$

In case 1. can I conclude that $f_n(x)\to f(x)$ as $n\to\infty$ for almost every $x\in \mathbb R^d$ ? What about case 2. ?

If not, for sure in case 2. almost-everywhere convergence holds true along a subsequence. Is this true also in case 1.?