Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations

Let (X,ℱ,µ) be a finite measure space and τ a null preserving transformation on (X,ℱ,µ). Functions in Lorentz spaces L(p,q) associated with the measure μ are considered for pointwise ergodic theorems. Necessary and sufficient conditions are given in order that for any f in L(p,q) the ergodic average ${\displaystyle n^{-1}{\sum }^{n-1}\_\{i=0\}f\circ {\tau }^i\left(x\right)}$ converges almost everywhere to a function f* in ${\displaystyle L(p_1,q_1]}$, where (pq) and ${\displaystyle (p_1,q_1]}$ are assumed to be in the set ${\displaystyle \{(r,s):r=s=1,\text{or}1<r<\infty \text{and}1\le s\le \infty ,\text{or}r=s=\infty \}}$. Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized and unified