Counter-examples to the Dunford–Schwartz pointwise ergodic theorem on $$varvec{L^1+L^infty }$$ L 1 + L ∞

Extending a result by Chilin and Litvinov, we show by construction that given any $$sigma $$ -finite infinite measure space $$(Omega ,mathcal {A}, mu )$$ and a function $$fin L^1(Omega )+L^infty (Omega )$$ with $$mu ({|f|>varepsilon })=infty $$ for some $$varepsilon >0$$ , there exists a Dunford–Schwartz operator T over $$(Omega ,mathcal {A}, mu )$$ such that $$frac{1}{N}sum _{n=1}^N (T^nf)(x)$$ fails to converge for almost every $$xin Omega $$ . In addition, for each operator we construct, the set of functions for which pointwise convergence fails almost everywhere is residual in $$L^1(Omega )+L^infty (Omega )$$ .