Several Integrable Combinations

Let $f, g : [a, b] \to \mathbb R$ be Riemann-integrable.

Problem 1. Prove that the following functions are Riemann-integrable:

$f + g : [a, b]\to \mathbb R$ ,
$cf : [a, b]\to \mathbb R$ for any $c \in \mathbb R$ .

Solution. We prove the first result. Fix $\epsilon > 0$ . For any $k_1, k_2 > 0$ , find partitions $P, Q \subseteq [a, b]$ with $R := P \cup Q$ such that

$\begin{aligned} U(f, R) - L(f, R) &< k_1 \cdot \epsilon \\ U(g, R) - L(g, R) &< k_2 \cdot \epsilon \end{aligned}$

By definition, on each $[x_{i-1}, x_i]$ ,

$\begin{aligned} m_i(f, R) \leq f \leq M_i(f, R) \\ m_i(g, R) \leq g \leq M_i(g, R) \end{aligned}$

Adding yields

$m_i(f, R) + m_i(g, R) \leq f+g \leq M_i(f,R) + M_i(g,R).$

Taking supremums and infimums yields

$\begin{aligned} m_i(f, R) + m_i(g, R) \leq m_i(f+g,R) \\ M_i(f+g,R) \leq M_i(f,R) + M_i(g,R) \end{aligned}$

Thus,

$\displaystyle \begin{aligned} & \sum_{i=1}^n (M_i(f+g,R) - m_i(f+g,R)) \\ &\leq (U(f, R) - L(f, R)) + (U(g, R) - L(g, R)) \\ &< (k_1 + k_2) \cdot \epsilon. \end{aligned}$

Setting $k_1 = k_2 = 1/2$ yields the desired result. We leave it as a bookkeeping exercise to verify that

$\displaystyle \int_a^b (f+g) = \int_a^b f + \int_a^b g.$

For the second result, the result is obvious with $c = 0$ . Follow a similar proof for $c > 0$ via the estimate

$M_i(cf, P) - m_i(cf, P) = c(M_i(f, P) - m_i(f, P)).$

For $c < 0$ , write $cf = -1 \cdot (-c)f$ . It suffices to establish the case $c = -1$ . The proof is almost identical with the exception that

$M_i(-f, P) = -m_i(f, P),\quad m_i(-f, P) = - M_i(f, P)$

so that

$M_i(-f, P) - m_i(-f, P) = M_i(f, P) - m_i(f, P).$

Finally we leave it as a bookkeeping exercise to verify that

$\displaystyle \int_a^b (cf) = c \int_a^b f.$

Problem 2. Prove that the following functions are Riemann-integrable:

$f^2 : [a, b] \to \mathbb R$ ,
$fg : [a, b] \to \mathbb R$ ,
$|f| : [a, b] \to \mathbb R$ ,
$f^+ := \max\{0, f\} : [a, b] \to \mathbb R$ ,
$\max\{f, g\} : [a, b] \to \mathbb R$ .

Remarkably, closed forms for these functions in terms of $\displaystyle \int_a^b f, \int_a^b g$ are highly nontrivial.

Solution. We will establish these properties in sequence.

Since $(\cdot)^2 : [0, \max f([a,b])] \to \mathbb R$ is continuous, $f^2 = (\cdot)^2 \circ f$ is Riemann-integrable.
Then $fg = \frac 12 ((f+g)^2 - f^2 - g^2)$ is Riemann-integrable.
Furthermore, $\sqrt{\cdot} : [0, \max |f([a,b])|] \to \mathbb R$ is continuous, $|f| = \sqrt{\cdot} \circ f^2$ is Riemann-integrable.
Then $f^+ = \frac 12 (f + |f|)$ is Riemann-integrable.
Finally, $\max\{f, g\} = f + \max\{0, g-f\}$ is Riemann-integrable.

Problem 3. Prove that $\displaystyle \left| \int_a^b f \right| \leq \int_a^b \left| f \right|$ .

Solution. Apply the monotonicity of integration to $0 \leq f + |f| \leq 2|f|$ .

Remark. If $f,g$ are Lebesgue-integrable implies that the following functions are Lebesgue-integrable:

$f + g : [a, b]\to \mathbb R$ ,
$cf : [a, b]\to \mathbb R$ for any $c \in \mathbb R$ ,
$\phi \circ f : [a, b] \to \mathbb R$ , where $\phi$ is continuous,

then all the functions in Problem 2 will also be Lebesgue-integrable, and the estimate in Problem 3 still holds. In other words, Problem 1 together with closure under composition with a continuous function, if both still hold when “Riemann-integrable” is replaced with “Lebesgue-integrable” (more generally, any suitable predicate $\Phi$ on the set of functions $[a, b] \to \mathbb R$ ) will imply the desired Lebesgue-integrability properties in Problem 2 and Problem 3.

—Joel Kindiak, 21 Jan 25, 1736H

KindiakMath

Several Integrable Combinations

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Several Integrable Combinations

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