Linearity and Monotonicity of the Riemann Integral
Important properties of the Riemann integral.
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Study Linearity and Monotonicity of the Riemann Integral
The integral can be seen as a mapping from the space \(\mathcal{T}([a,b])\) to \(\mathbb{R}\). In the literature, mappings from vector spaces to a field (in this case \(\mathbb{R}\)) are called functionals. The following result shows that the integral is a linear and monotone functional.
Theorem 1. \(\int_a^b:\mathcal{T}([a,b])\to\mathbb{R}\) is linear and monotonic, that is, for all \(\phi,\psi\in\mathcal{T}([a,b])\) and all \(\lambda,\mu\in\mathbb{R}\) holds
\[\int_a^b \Big(\lambda\phi(x)+\mu\psi(x)\Big)\, dx=\lambda\int_a^b\phi(x)\, dx+\mu\int_a^b\psi(x)\, dx\]
if \(\phi(x)\leq\psi(x)\) for all \(x\in[a,b]\) (\(\phi\leq\psi\)) , then \[\int_a^b\phi(x)\, dx\leq\int_a^b\psi(x)\, dx.\]
Proof: This directly follows by the definition of the integral.\(\Box\)
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