Convergence in measure and convergence in $L^p$

Question

If $f_n$ is convergent to $f$ in measure and $\|f_{n}(x)\|_{L^{p}(\mathbb{R})}=\|f(x)\|_{L^{p}(\mathbb{R})}$.

Does it implies that $f_n$ is convergent in $L^p$?

Answer 1

Suppose not: then for some $\delta$ and some $n_k\uparrow\infty$, $$\lVert f_{n_k}-f\rVert_p\geqslant\delta,\quad k\geqslant 1.$$ Using the definition of convergence in measure, we can construct $m_k\uparrow\infty$ such that $\lambda\{x, |f_{n_{m_k}}(x)-f(x)|>2^{-k}\}\leqslant 2^{-k}.$

The sequence $(f_{n_{m_k}})_{k\geqslant 1}$ converges almost everywhere to $f$ and $\lVert f_{n_{m_k}}\rVert_p\to\lVert f\rVert_p$, hence $\lVert f_{n_{m_k}}-f\rVert_p\to 0$, a contradiction.