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I am following the proof of approximating point-wise, non-negative measurable functions by simple functions in Stein and Shakarchi (2009).

I am little confused in their initial steps of truncation.

They state:

For $k\geq1$, let $Q_k$ denote the cube centered at the origin with a side length $k$. Then we define, $$F_k(x)=f(x) \space\space\text{if $x\in Q_k,f(x)\leq k$},\\k\space\space\text{if $x\in Q_k,f(x)> k$}\\0\space\space \text{otherwise}.$$ $F_k(x)\rightarrow f(x)$ as $k\rightarrow\infty$ for all $x$.

My questions:

(1) Why are we defining $F_k(x)$ here. Is this a simple function? Just recalling the definition, I thought the simple function is a finite sum of characteristic functions over a measurable set. I don't see how $F$ is related.

(2) Why does $F_k$ converge point-wise to $f$?

Reference: $\textit{Real Analysis: Measure Theory, Integration, and Hilbert Spaces}$. Elias M. Stein, Rami Shakarchi. Princeton University Press, 2009.

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1 Answer 1

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It looks to me like it is a simple function outside of the $k$-cube, and it is a bounded, continuous function inside the $k$-cube - is that good enough for the proof they are doing?

It does converge pointwise to $x$, because for each $x$, eventually both of these is true: 1.) $x$ is in every $k$-cube for every $k $ above some $N_1$, and $f(x) < k$ for every $k$ above some $ N_2$. So, for each $x$, eventually $F_k(x) = f(x)$.

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