If $f,g$ are integrable on $[a,b]$ then so are $\lvert f\rvert$, $f^{2}$, $fg$, $\max(f,g)$ and $\min(f,g)$, and $\bigl\lvert\int_a^b f\bigr\rvert \le \int_a^b\lvert f\rvert$

Let $a < b$ be reals and let $f, g : [a,b] \to \mathbb{R}$ be integrable (The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as $\sup_P L(f,P)$ and $\inf_P U(f,P)$, Darboux integrability as their equality, and the notation $\int_a^b f$). Then:

1. $|f|$, $f^{2}$ and $fg$ are integrable on $[a,b]$ (Absolute value in an ordered field, Integer powers $a^m$);
2. $\max(f,g)$ and $\min(f,g)$, defined pointwise (Maximum and minimum of a set), are integrable on $[a,b]$;
3. the triangle inequality for the integral: $\Bigl|\int_a^b f\Bigr| \;\le\; \int_a^b |f| .$

Claim 3 is stated with $a < b$ and is not orientation-invariant. For $u > v$ the right-hand side is $-\int_v^u |f| \le 0$ while the left-hand side is $\ge 0$, so the inequality as written is false there. The form valid for every pair $u,v$ on which $f$ is integrable (The integral with oriented limits: $\int_a^a f := 0$ and $\int_b^a f := -\int_a^b f$) is

$\Bigl|\int_u^v f\Bigr| \;\le\; \Bigl|\int_u^v |f|\,\Bigr| ,$

and that is the form the estimates below on this page use whenever the limits are not known to be in increasing order.

The converse of claim 1 fails. Integrability of $|f|$ does not give integrability of $f$; the witness is on the companion page.

• Every integrability clause comes from one theorem plus linearity. The only input that produces integrability is If $f$ is integrable on $[a,b]$ with values in $[m,M]$ and $\varphi$ is continuous on $[m,M]$, then $\varphi \circ f$ is integrable, with Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and $\int_a^b(\lambda f+\mu g) = \lambda\int_a^b f + \mu\int_a^b g$ recombining the pieces; claim 3 additionally uses If $f \le g$ on $[a,b]$ and both are integrable then $\int_a^b f \le \int_a^b g$; and $m(b-a) \le \int_a^b f \le M(b-a)$, which is the one place an inequality between integrals is needed. The identities of [L5] are algebra, and they are what turns a statement about composing with $|\cdot|$ and $(\cdot)^2$ into statements about products and lattice operations. In particular no new estimate on Darboux sums is made here.

• The polarisation identity is used, and it is why $f^2$ comes first. There is no direct route from integrability of $f$ and of $g$ to integrability of $fg$ through the composition theorem, because $(s,t) \mapsto st$ is a function of two variables and the theorem composes with one. Writing $fg$ through squares of sums and differences reduces it to the one-variable case.

• The inequality of claim 3 is the integral analogue of the triangle inequality, and like it, it can be strict: for $f(x) = x$ on $[-1,1]$ the left-hand side is $0$ and the right-hand side is $1$.

• Forward reference, orientation only. The witness refuting the converse of claim 1 is A function that is not Riemann integrable although $\lvert f\rvert$ is ↗ on the companion page; nothing above depends on it.