## ABSTRACT

In what follows we let R + = { x ∈ R : x ≥ 0 } ${\mathbb{R}}^{ + } = \{ x \in {\mathbb{R}}:\text{ }x \ge 0\}$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351056823/5c92c9a4-ddd5-4d42-82bc-8e5903ddaaca/content/inline-math2_1.tif"/> . We let R ^ $\hat{\mathbb{R}}$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351056823/5c92c9a4-ddd5-4d42-82bc-8e5903ddaaca/content/inline-math2_2.tif"/> denote the extended reals, and we let R ^ = { x ∈ R ^ : x ≥ 0 } . $\hat{\mathbb{R}} = \{ x \in \hat{\mathbb{R}} : x \ge 0\} .$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351056823/5c92c9a4-ddd5-4d42-82bc-8e5903ddaaca/content/inline-math2_3.tif"/> . Thus R ^ + = R + ∪ { + ∞ } . $\hat{\mathbb{R}}^{ + } = {\mathbb{R}}^{ + } \mathop \cup \nolimits \{ + \infty \} .$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351056823/5c92c9a4-ddd5-4d42-82bc-8e5903ddaaca/content/inline-math2_4.tif"/>