Let $f:[0,1]\times[0,1] \rightarrow \mathbb{R}$ be given with $f(x)=\frac{1}{||x||}$ for every $x≠(0,0)$.
Construct a sequence of step functions $(f_k)_{k\in\mathbb{N}}\in T^{inc}$, so that $f_k\rightarrow f$ a.e. as $k\rightarrow\infty$, and show that the limit $\lim_{k\rightarrow\infty}\int f_k$ exists and it's bounded.
Here $T^{inc}$ is the set of all increasing step functions.
I'm not very good at constructing step functions, but I know that for a function $f:(0,1]\rightarrow\mathbb{R}$ with $f(x)=\frac1x$, a step function $f_k\in T^{inc}$ that converges to $f$ as $k\rightarrow\infty$ would be $f_k=\sum^{2^k}_{j=1}\frac{2^k}j\mathbb{1}_{((j-1)/2^k,j/2^k]}$
Do you think I could use the same step function? Or do you any advice how to proceed or if there is some usual procediment in constructing step fucntion?