Let D be a bounded subset of ℝ and f: D →ℝ be a bounded function. Suppose D ⊆[a, b] for a, b ∈ℝ

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Let D be a bounded subset of ℝ and f: D →ℝ be a bounded...

Let $D$ be a bounded subset of $\mathbb{R}$ and $f: D \rightarrow \mathbb{R}$ be a bounded function. Suppose $D \subseteq[a, b]$ for $a, b \in \mathbb{R}$ and $f^{*}:[a, b] \rightarrow \mathbb{R}$ is defined by $$f^{*}(x):=\left\{\begin{array}{ll} f(x) & \text { if } x \in D \\ 0 & \text { otherwise } \end{array}\right.$$ The function $f$ is said to be integrable (on $D)$ if the function $f^{*}$ is integrable (on $[a, b])$. In this case, we define the Riemann integral of $f$ $($ on $D)$ by $$ \int_{D} f(x) d x:=\int_{a}^{b} f^{*}(x) d x $$ (i) Show that the above definition is independent of the interval $[a, b]$ containing $D$. (ii) Show that analogues of Propositions $6.15$ and $6.18$ hold for integrable functions on $D$.

Let $D$ be a bounded subset of $\mathbb{R}$ and $f: D \rightarrow \mathbb{R}$ be a bounded function. Suppose $D \subseteq[a, b]$ for $a, b \in \mathbb{R}$ and $f^{*}:[a, b] \rightarrow \mathbb{R}$ is defined by
$$f^{*}(x):=\left\{\begin{array}{ll}
f(x) & \text { if } x \in D \\
0 & \text { otherwise }
\end{array}\right.$$
The function $f$ is said to be integrable (on $D)$ if the function $f^{*}$ is integrable (on $[a, b])$. In this case, we define the Riemann integral of $f$ $($ on $D)$ by
$$
\int_{D} f(x) d x:=\int_{a}^{b} f^{*}(x) d x
$$
(i) Show that the above definition is independent of the interval $[a, b]$ containing $D$.
(ii) Show that analogues of Propositions $6.15$ and $6.18$ hold for integrable functions on $D$.

Key Concepts

Riemann Integration

Riemann integration is the process of defining the area under a curve for a bounded function over a closed interval. The classical approach partitions the interval into subintervals, computes sums of function values times subinterval lengths, and takes a limit as the maximum subinterval length tends to zero. It forms the foundation for understanding integration in elementary real analysis and connects geometric intuition with analytic precision.

Extension of Functions by Zero

Extending a function by zero involves defining a new function on a larger domain in which the original function is preserved on its original domain and is set to zero elsewhere. This method allows one to study the properties of functions defined on arbitrary (possibly non-interval) sets by embedding them into a standard setting, such as a closed interval, and ensures that the integral over the larger domain captures the contribution from the original function without interference from the added parts.

Independence of the Domain of Integration

The concept of independence of the domain in integration addresses the idea that the value of the integral does not depend on the particular choice of the interval used to contain the domain of the original function. When a function is extended by zero outside its domain, any additional regions (where the function is zero) do not affect the overall integral, provided the integration is performed over any interval that covers the original set. This concept ensures well-definedness and consistency in the definition of integrals on subsets.

Properties of Integrable Functions and Their Analogues

Key properties of integrable functions, such as linearity, additivity, and order preservation, extend naturally when considering functions defined on arbitrary bounded subsets by means of their zero extension. The analogues of standard propositions (such as those concerning the effect of modifying a function on a set of measure zero or the sum of integrals over disjoint regions) hold true in this generalized context, ensuring that the fundamental algebraic and analytic characteristics of the Riemann integral are preserved.

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