Something which I am not sure how is it inferred.
On page 307 of the book Ergodic Theorems by Ulrich Krengel, they write that: " $M_l(x) = \mu(\{ z : k_1(x,z)\geq 1/l \} )$. Let $l(x)$ be the smallest integer with $M_l(x)\geq 3/4$ . Put $a(x)=l(x)/2$. Then $a(\cdot)$ is strictly positive and the measurable set $A_x =\{ k_1(x,\cdot) \geq a(x) \} $ has measure $\mu(A_x)\ge 3/4$."
where $k_1$ is a stricly positive everywhere integral kernel.
I don't understnad why $\mu(A_x)\ge 3/4$, I mean from what did they infer this? I mean $k_1(x,z)\geq l(x)/2$ doesn't mean that $k_1(x,z)\geq 1/l(x)$ unless $l(x)>1$; Perhaps they meant $a(x)= 2/l(x)$, in which case I can see how they inferred this.
We can start off by calculating: $$\mu(A_x) = \mu(\{k_1(x, \cdot) \geq a(x)\}) = \mu(\{z : k_1(x, z) \geq a(x)\}) = M_{a(x)}(x)$$
Now $a(x)$ has been chosen less than $l(x)$ and $M_l(x) \geq 3/4$. So we are done if $M_s(x) \geq M_t(x)$ for all $s \leq t$. This is true because the set $\{z : k_1(x, z) \geq s\}$ is a superset of $\{z : k_1(x, z) \geq t\}$.