Riesz's Convergence Theorem

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Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $p \in \R$, $p \ge 1$.

Let $\sequence {f_n}_{n \mathop \in \N}, f_n: X \to \R$ be a sequence in Lebesgue $p$-space $\map {\LL^p} \mu$.


Suppose that the pointwise limit $f := \displaystyle \lim_{n \mathop \to \infty} f_n$ exists $\mu$-almost everywhere, and that $f \in \map {\LL^p} \mu$.

Then the following are equivalent:

$(1): \quad \displaystyle \lim_{n \mathop \to \infty} \norm {f - f_n}_p = 0$
$(2): \quad \displaystyle \lim_{n \mathop \to \infty} \norm {f_n}_p = \norm f_p$

where $\norm {\, \cdot \,}_p$ denotes the $p$-seminorm.


Proof


Source of Name

This entry was named for Frigyes Riesz.


Sources