📂Lebesgue Spaces

Lp Convergence

Definition ¹

If a sequence of functions $\left\{ f_{n} \right\}_{n \in \mathbb{N}}$ satisfies the following for some function $f$, then $\left\{ f_{n} \right\}$ is said to converge to $f$ in $L^{p}$.

$$ \lim_{n \to \infty} \left\| f_{n} - f \right\|_{p} = 0 $$

The sequence $\left\{ f_{n} \right\}_{n \in \mathbb{N}}$ is said to be Cauchy in $L^{p}$ if it satisfies the following.

$$ \lim_{n, m \to \infty} \left\| f_{n} - f_{m} \right\|_{p} = 0 $$

Explanation

Of course, $\left\| \cdot \right\|_{p}$ is defined as the following with respect to $p$-norm.

$$ \left\| f \right\|_{p} := \left( \int_{E} | f |^{p} dm \right) ^{{{1} \over {p}}} $$

The statement that a sequence of functions converges in $L^{p}$ means convergence in the sense of norm. In the properties of Lebesgue space, when $p \le q$, if $f_{n}$ converges in $L^{q}$, it means convergence in $L^{p}$.

See Also

• Convergence in $L^{p}$ $\implies$ Measure Convergence