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I am a beginner of measure theory. After studying the measurable function from Stein Shakarchi's book I find some approximation theorems of measurable functions by a sequence of simple and step functions. I have read them all. But still I am struggling in it. What is the way of approximating a given measurable function by a sequence of simple and step functions is still unknown to me. I also don't find any online resource which elaborates this fact nicely by demonstrating some useful examples.

So can anybody suggest me some books which have nice explanation in this area and also which are available online? Then it will be really helpful for me.

Thank you very much.

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One of the reasons, why the approximation by simple functions is useful, is that it gives a natural way to define the integral of measurable functions.

Let $\mu$ be a measure on a measurable space $(\Omega,\mathcal{F})$. If we are interested in definining an integral operator $I$, then this operator should have two important properties:

  • linearity, i.e. $$I(\alpha f + \beta g) = \alpha I(f) + \beta I(g) \tag{1}$$ for any constants $\alpha,\beta \in \mathbb{R}$ and "integrable" functions $f,g$. (Abstractly speaking, "integrable" means for the moment being that $f$ and $g$ are in the domain of the integral operator.)
  • monotonicity, i.e. $$f \leq g \implies I(f) \leq I(g) \tag{2}$$ for any integrable $f,g$.

The property of monotonicity is natural insofar as we would like to interprete the integral of a function as the "area under the curve".

Now if we are given a simple function of the form

$$f(x) = \sum_{j=1}^k c_j 1_{A_j}(x)$$

for some constants $c_j \geq 0$ and measurable sets $A_j \in \mathcal{A}$, then the linearity of the integral operator gives

$$I(f) = \sum_{j=1}^k c_j I(1_{A_j}).$$

Since we already know how to measure $A_j$ (and hence $1_{A_j}$) the only natural way is to put

$$I(1_A) = \mu(A) \qquad \text{for all} \,\, A \in \mathcal{A}. \tag{3}$$

Hence,

$$I(f) = \sum_{j=1}^k c_j \mu(A_j).$$

This means that we have defined the integral for non-negative simple functions. The next step is to extend the integral operator to non-negative measurable functions, and here the approximation by simple functions plays a key role. For a measurable function $f \geq 0$ we can find a sequence of simple functions $(f_n)_{n \in \mathbb{N}}$, $f_n \geq 0$, such that $f_n \uparrow f$. Because of the monotonicity of the integral operator, we find that $I(f_n)$ is an increasing sequence and it is natural to define

$$I(f) := \sup_{n \in \mathbb{N}} I(f_n).$$

Thanks to the linearity of the integral we can get rid of the assumption on the non-negativity of $f$. Indeed: Writing $f=f^+-f^-$ we define

$$I(f) := I(f^+)- I(f^-)$$

whenever the right-hand side makes sense, i.e. $I(f^+)< \infty$ and/or $I(f^-)<\infty$.

Thus, we have successfully defined in a natural way an integral operator for a large class of functions $f$. They key tools are $(1)$, $(2)$ and the approximation by step functions.

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  • $\begingroup$ Would the downvoter care to comment? Thanks... $\endgroup$ – saz May 7 '18 at 13:44

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