Convergence in $L^p$

A convergence in $L^p$ is norm convergence in a Lp space . Let $(X,\mathcal{A},\mu)$ be a measure space and let $1\le p<\infty$. A sequence $(f_n)$ in $L^p(X)$ converges in $L^p$ to $f\in L^p(X)$ if

where $\|\cdot\|_p$ is the Lp norm . For $p=\infty$, one defines convergence in $L^\infty$ by $\|f_n-f\|_\infty\to 0$, where $\|\cdot\|_\infty$ is the essential supremum norm (see essential supremum ).

Convergence in $L^p$ controls the size of the error in an averaged sense. In particular, for $1\le p<\infty$ the estimate

shows that convergence in $L^p$ implies convergence in measure .

Examples:

• On $([0,1],\mathcal{B},\lambda)$, the functions $f_n(x)=x^n$ satisfy $f_n\to 0$ in $L^p$ for every $1\le p<\infty$ since

  $$\|f_n\|_p^p=\int_0^1 x^{np}\,dx=\frac{1}{np+1}\to 0.$$

• On $([0,1],\mathcal{B},\lambda)$ and for fixed $1\le p<\infty$, the functions $g_n = n^{1/p}\mathbf{1}_{[0,1/n]}$ satisfy $g_n\to 0$ in measure, but not in $L^p$, because

  $$\|g_n\|_p^p = \int_0^{1/n} n\,dx = 1$$

  for all $n$.