3.10. Additive and Linearity Properties of Upper and Lower Integrals

Proposition 3.3

Let \(c\in (a, b)\). We have the following identities for upper and lower integrals:

\[\overline{\int_a^b} f \dif\alpha = \overline{\int_a^c} f \dif\alpha + \overline{\int_c^b} f \dif\alpha\]

and

\[\underline{\int_a^b} f \dif\alpha = \underline{\int_a^c} f \dif\alpha + \underline{\int_c^b} f \dif\alpha\]

Proof. We only prove the identity for upper integrals. We will prove this identity by showing both LHS \(\geq\) RHS and LHS \(\leq\) RHS.

\noindent Proof of LHS \(\geq\) RHS: Let \(P_1\) and \(P_2\) be partitions of \([a, c]\) and \([c, b]\), respectively. Let \(P = P_1 \cup P_2\). We have

\[U(P, f, \alpha) = U(P_1, f, \alpha) + U(P_2, f, \alpha) \geq\overline{\int_a^c} f \dif\alpha + \overline{\int_c^b} f \dif\alpha\]

Taking the infimum over \(P\) yields

\[\overline{\int_a^b} f \dif\alpha = \inf_{P \in \CALP[a, b]} U(P, f, \alpha) \geq\overline{\int_a^c} f \dif\alpha + \overline{\int_c^b} f \dif\alpha\]

\noindent Proof of LHS \(\leq\) RHS: Let \(\varepsilon > 0\) be arbitrary. There exist a partition \(P_1\) of \([a, c]\) and a partition \(P_2\) of \([c, b]\) such that

\[\begin{split}U(P_1, f, \alpha) - \varepsilon / 2 & < \overline{\int_a^c} f \dif\alpha\\ U(P_2, f, \alpha) - \varepsilon / 2 & < \overline{\int_c^b} f \dif\alpha\end{split}\]

Let \(P = P_1 \cup P_2\). Adding the above two inequalities yields

\[U(P, f, \alpha) - \varepsilon = U(P_1, f, \alpha) - \varepsilon/2 + U(P_2, f, \alpha) - \varepsilon/2 < \overline{\int_a^c} f \dif\alpha + \overline{\int_c^b} f \dif\alpha\]

Taking the infimum over \(P\) on both sides yields

\[\overline{\int_a^b} f \dif\alpha - \varepsilon = \inf_{P \in \CALP[a, b]}(U(P, f, \alpha) - \varepsilon) \leq\overline{\int_a^c} f \dif\alpha + \overline{\int_c^b} f \dif\alpha\]

Since

\[\overline{\int_a^b} f \dif\alpha - \varepsilon\leq\overline{\int_a^c} f \dif\alpha + \overline{\int_c^b} f \dif\alpha\]

holds for every \(\varepsilon > 0\),

\[\overline{\int_a^b} f \dif\alpha\leq\overline{\int_a^c} f \dif\alpha + \overline{\int_c^b} f \dif\alpha\]

Proposition 3.4

We have the following inequalities about upper and lower integrals of sums of two functions:

(3.27)\[\overline{\int_a^b} f+g \dif\alpha\leq\overline{\int_a^b} f \dif\alpha + \overline{\int_a^b} g \dif\alpha\]

and

\[\underline{\int_a^b} f+g \dif\alpha\geq\underline{\int_a^b} f \dif\alpha + \underline{\int_a^b} g \dif\alpha\]

Proof. We only prove (3.27). Let \(P\) be any partition of \([a, b]\). On each interval \([x_{k-1}, x_k]\), we have

\[f(x) + g(x) \leq\sup_{x \in [x_{k-1}, x_k]} f(x) + \sup_{x \in [x_{k-1}, x_k]} g(x) \quad\forall x \in[x_{k-1}, x_k]\]

Taking the supremum on both sides over \(x\) yields

\[\sup_{x \in [x_{k-1}, x_k]} f(x) + g(x) \leq\sup_{x \in [x_{k-1}, x_k]} f(x) + \sup_{x \in [x_{k-1}, x_k]} g(x) \]

Then summing over \(k\):

\[\begin{split}U(P, f+g, \alpha) & = \sum_{k}\sup_{x \in [x_{k-1}, x_k]} f(x) + g(x) \\& \leq\sum_{k}\sup_{x \in [x_{k-1}, x_k]} f(x) + \sum_{k}\sup_{x \in [x_{k-1}, x_k]} g(x) \\& = U(P, f, \alpha) + U(P, g, \alpha) \end{split}\]

Finally, the inequality is proved by taking the infimum on both sides over \(P\).

Example 3.3

The inequality (3.27) may be strict. Consider \(f(x) = \ind_{\Q} (x)\) and \(g(x) = -\ind_{\Q}(x)\) restricted on \([a, b]\). We have

\[\overline{\int_a^b} f+g \dif\alpha = \overline{\int_a^b} 0 \dif\alpha = 0 \]

and

\[\overline{\int_a^b} f \dif\alpha = b-a \quad\text{and}\quad\overline{\int_a^b} g \dif\alpha = 0 \]