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The Banach-Saks property is an important tool in analysis with applicationsranging from partial differential equations (PDEs) to calculus of variationsand probability theory. We survey the Banach-Saks property for $L^p$-spaces,with a particular emphasis on the case where $p=1$. In other words, we revisitthe celebrated result by W. Szlenk (1965) in a more general context,demonstrating that $L^1$-spaces possess the weak Banach-Saks property.

Banach-Saks Theorem for $L^1$ revisited