Measure Theory – Why Lebesgue Integral is Defined Through Simple Functions

lebesgue-integralmeasure-theory

I'm reading the Measure Theory and Integration chapter from Terrance Tao's Analysis-2. In the chapter of Lebesgue Integration (11), initially the integral is introduced for positive valued functions and then generalized to negative valued function by first defining a way to split up the negative and positive value of function.

What I don't understand is, why can't we directly integrate functions which take any value in the Lebesgue Integration? Could someone give an intuition to me on why we do this?

I understand stuff like defining the measure function, why we need measurable sets , measurable functions etc but this point I am confused as heck.

Edit : I found this relevant MO post
https://mathoverflow.net/q/25161/159957

Best Answer

You don't have to do it this way. This is just one (convenient) approach. It is convenient because doing it this way allows us to very easily handle $\infty$.

There are other textbooks (e.g Lang, Amann and Escher) which do things differently. Fix a measure space $(X,\mathfrak{M},\mu)$, and a real/complex finite-dimensional (for simplicity) normed vector space $V$. We always equip $V$ with the topology induced by the norm, and the Borel $\sigma$-algebra associated to this topology. Fun fact: since $V$ is finite-dimensional, all norms on $V$ generate the same topology, and hence the same Borel $\sigma$-algebra.

Approach 1: Directly Integrating $V$-valued functions.

Now we can define a $V$-valued simple function on $X$ in the obvious way, namely a measurable function $s:X\to V$ with a finite image set. Then, letting $v_1,\dots, v_k$ be the distinct elements of the image, we can decompose it as \begin{align} s=\sum_{i=1}^k\mathbf{1}_{s^{-1}(\{v_i\})}v_i, \end{align} i.e it is exactly what we expect: a finite combination of indicators. Then, we define $\int_Xs\,d\mu$ in the obvious way, provided each of the sets defining the indicators has finite measure. Then, we define $\mathcal{L}^1(X,\mu,V)$ to be the set of measurable $f:X\to V$ for which there is a "$L^1$-Cauchy-sequence of simple functions" which approximates; more precisely the requirement is that there is a sequence $\{s_n\}_{n=1}^{\infty}$ of $V$-valued measurable functions such that

  1. we have $s_n\to f$ pointwise $\mu$-a.e. (approximate $f$)
  2. For every $\epsilon>0$, there is an $N\in\Bbb{N}$ such that for all $n,m\geq N$, we have $\int_X\|s_n-s_m\|_V\,d\mu <\epsilon$. ($L^1$-Cauchy).

A note about condition 2: you obviously need to see it's well-defined in the following sense: you need to know that the space of simple functions forms a vector space, so that $s_n-s_m$ is again a $V$-valued simple function. Also, $\|s_n-s_m\|_V$ is a $[0,\infty)\subset\Bbb{R}$-valued simple function. THese are the sorts of basic stuff treated more systematically in textbooks which is why I'm glossing over them.

Condition 2 implies that $\{\int_Xs_n\,d\mu\}_{n=1}^{\infty}$ is a Cauchy-sequence in $V$, so by completeness of $V$, it follows that this sequence has a limit, and we define $\int_Xf\,d\mu:=\lim\limits_{n\to\infty}\int_Xs_n\,d\mu$. One final well-definition check is required: we need to show that if we take a different sequence $\{s_n'\}_{n=1}^{\infty}$ satisfying $(1)$ and $(2)$, then we get the same limit. This is again proven in textbooks.

So, the long story short is that we have directly defined integration of $V$-valued measurable functions, as limits of integrals of an approximating sequence of simple functions. This idea must be extremely familiar to you from Riemann integration as well: the Riemann sums play the role of the simple functions, and the full Riemann integral is defined via a careful limiting procedure.

This approach doesn't rely on the total order of $[0,\infty]$, and with some tweaks can even handle infinite-dimensional Banach spaces, so in that sense may be desirable.

Approach 2: A Hybrid.

However, as I said in the beginning, being able to handle $\infty$ is a very convenient thing, because it makes the 3 major limit theorems (MCT, Fatou's lemma, and DCT) extremely easy to state. For this reason, it is good to first define integration of $[0,\infty]$ valued functions. Then, rather than splitting things up into positive and negative parts, followed by real/imaginary parts, we can proceed as follows:

  • Define $\mathscr{L}^1(X,\mu,V)$ as the set of measurable $f:X\to V$ such that $\|f\|_1:=\int_X\|f\|_V\,d\mu<\infty$. Note that $f:X\to V$ is measurable and $\|\cdot\|_V:V\to [0,\infty)$ is continuous, so the composition $\|f\|_V:X\to [0,\infty)$ is measurable, so the above integral makes sense. Note that $\|\cdot\|_1$ defines a semi-norm on $\mathscr{L}^1(X,\mu,V)$, so it allows us to give a topology on $\mathscr{L}^1(X,\mu,V)$.
  • Define $\mathcal{S}(X,\mu,V)$ to be the space of all simple integrable functions.
  • Prove that $\mathscr{L}^1(X,\mu,V)$ is a vector space over the same field as $V$, and that $\mathcal{S}(X,\mu,V)$ is a subspace. In fact, the simple functions $\mathcal{S}(X,\mu,V)$ are dense in $\mathscr{L}^1(X,\mu,V)$, with respect to the above semi-norm.
  • The integrtion map $\int_X(\cdot)\,d\mu:\mathcal{S}(X,\mu,V)\to V$ is linear and continuous with respect to the semi-norm on the domain, and the usual norm on $V$, i.e $\|\int_Xs\,d\mu\|\leq \int_X\|s\|_V\,d\mu$. For simple functions, this is a very trivial consequence of the triangle inequality.
  • Finally, we show that the integration map has a unique continuous linear extension $\int_X(\cdot)\,d\mu:\mathscr{L}^1(X,\mu,V)\to V$.

I should remark that for a finite-dimensional vector space, this way of stating things is a little unnecessarily abstract (because once we define integrals for $[0,\infty]$ functions, by splitting into positive and negative parts, we can define it for all real functions, then by dealing with each component of $f$ relative to a basis on $V$, we can define integrability of $f$), but for infinite-dimensional (separable) Banach spaces, this is necessary.

Anyway, I'm skipping over all proofs here, and glossing over several details because it's too much to write in an MSE post. But regardless of the "implementation details", the basic idea is to first define integrals for simple functions, and then extend by a limiting argument, the integral for other functions.

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