Counter-examples to the Dunford-Schwartz pointwise ergodic theorem on $L^1+L^{\mathrm{\infty }}$

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