Exercise 1.3.14 (Uniqueness of the Lebesgue integral)

Recall that (ii) implies $\int (f-g)=\int \; <!--\; nolimits\; -->f-\int \; <!--\; nolimits\; -->g$ whenever $f\ge g$ almost everywhere. Also, we can immediate yet another few useful properties:

(i)

(Monotonicity) If $f\le g$ then ${\int }_{{ℝ}^d}f\le {\int \; <!--\; nolimits\; -->}_{{ℝ}^d}g$.derived using (ii) by observing $\int g=\int \; <!--\; nolimits\; -->f+(g-f)=\int \; <!--\; nolimits\; -->f+\int \; <!--\; nolimits\; -->g-f\ge \int \; <!--\; nolimits\; -->f$.

(ii)

(Divisibility) If $E\subseteq {ℝ}^d$ is a Lebesgue measurable set, then ${\int }_{{ℝ}^d}f={\int \; <!--\; nolimits\; -->}_{{ℝ}^d}f\cdot 1_E+{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}f\cdot 1_{{ℝ}^d∕E}$follows directly from (ii) by noting that $f=f\cdot 1_E+f\cdot 1_{{ℝ}^d∕E}$.

(iii)

(Equivalence) If $f=g$ almost everywhere, then ${\int }_{{ℝ}^d}f={\int \; <!--\; nolimits\; -->}_{{ℝ}^d}g$.
Let $E$ be the null set on which $f\ne g$. We then have $\int f\cdot 1_E\le \int \; <!--\; nolimits\; -->\sup <!--\; FUNCTION\; APPLICATION\; -->f\cdot 1_E=\mathrm{Simp}{\mathrm{\int \; <!--\; nolimits\; -->}<!--\; FUNCTION\; APPLICATION\; -->}_{{ℝ}^d}\sup <!--\; FUNCTION\; APPLICATION\; -->f\cdot 1_E=\sup <!--\; FUNCTION\; APPLICATION\; -->f\cdot \mathrm{Simp}{\mathrm{\int \; <!--\; nolimits\; -->}<!--\; FUNCTION\; APPLICATION\; -->}_{{ℝ}^d}{1}_E=\sup <!--\; FUNCTION\; APPLICATION\; -->f\cdot m(E)=\sup <!--\; FUNCTION\; APPLICATION\; -->f\cdot 0=0$. Similarly $\int g\cdot 1_E=0$. But we also have by additivity (ii) that $\int f=\int \; <!--\; nolimits\; -->f\cdot 1_E+\int \; <!--\; nolimits\; -->f\cdot 1_{{ℝ}^d∕E}=\int \; <!--\; nolimits\; -->f\cdot 1_{{ℝ}^d∕E}=\int \; <!--\; nolimits\; -->g\cdot 1_{{ℝ}^d∕E}=\int \; <!--\; nolimits\; -->g\cdot 1_{{ℝ}^d∕E}+\int \; <!--\; nolimits\; -->g\cdot 1_{{ℝ}^d∕E}=\int \; <!--\; nolimits\; -->g$.

Now we somehow need to connect the area where two integrals $\int$ and ${\int }^{\prime }$ agree (i.e., simple functions) with the rest of the domain (i.e., Lebesgue measurable functions). We will try to construct a sequence ${(h_n)}_{n\in \mathbb{N}}$ such that $\int f=\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}\mathrm{Simp}{\mathrm{\int \; <!--\; nolimits\; -->}<!--\; FUNCTION\; APPLICATION\; -->}_{{ℝ}^d}{h}_n$. That way the value of $\int$ will be always dependent on the value of $\mathrm{Simp}{\int }_{{ℝ}^d}$, and thus unique.

Pick an arbitrary $𝜖>0$. By properties (iii) and (iv) we can find a $n\in \mathbb{N}$ such that

The key observation at this point (somehow I missed this for a whole day) is that $\min <!--\; FUNCTION\; APPLICATION\; -->\{f,n\}$ is a bounded Lebesgue measurable function. By Exercise 1.3.4 we thus can find a sequence of simple functions which uniformly converges to $\min <!--\; FUNCTION\; APPLICATION\; -->\{f,n\}$ and is increasing. In other words, for the given $𝜖$ we have a simple function $h$ such that $d_{\infty }\left(\min <!--\; FUNCTION\; APPLICATION\; -->\{f,n\}\cdot 1_{B(0,n)},h\right)\le 𝜖∕(2\cdot m(B(0,n)))$. The big difference is that the uniform convergence (unlike pointwise convergence) allows us to draw conclusions about the corresponding integrals using the few properties given in the theorem:

We thus conclude

Since our choice of $𝜖>0$ was arbitrary, we can easily construct a sequence ${(h_n)}_{n\in \mathbb{N}}$ of increasing simple functions whose simple integrals converge to $\int f$. Notice that we also have the pointwise convergence of ${(h_n)}_{n\to \infty }$ to $f$. This is very important, since for any other integral ${\int }^{\prime }$ satisfying the above properties, we can construct a similar sequence ${(h_n^{\prime })}_{n\in \mathbb{N}}\to f$ using the steps above, and the result will be

since for finite simple functions pointwise convergence $\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}{h}_n=\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}{h}_n^{\prime }$ necessarily implies uniform convergence, and thus convergence in measure $\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}\mathrm{\int \; <!--\; nolimits\; -->}<!--\; FUNCTION\; APPLICATION\; -->h_n=\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}\mathrm{\int \; <!--\; nolimits\; -->}<!--\; FUNCTION\; APPLICATION\; -->h_n^{\prime }$.