Diamond integral

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The $\Diamond$ integral was introduced to fix the issues with the $\Diamond_{\alpha}$ integral.

Properties

Theorem: The following formula holds: $$\int_a^a f(t) \Diamond t=0.$$

Proof:

Theorem: The following formula holds: $$\int_a^b f(t) \Diamond t= \int_a^c f(t) \Diamond t + \int_c^b f(t) \Diamond t.$$

Proof:

Theorem: The following formula holds: $$\int_a^b f(t) \Diamond t= -\int_b^a f(t) \Diamond t.$$

Proof:

Theorem (Sum Rule): The following formula holds: $$\int_a^b (f+g)(t) \Diamond t= \int_a^b f(t) \Diamond t + \int_a^b g(t) \Diamond t.$$

Proof:

Theorem (Constant Multiple): The following formula holds: $$\int_a^b \alpha f(t) \Diamond t= \alpha \int_a^b f(t) \Diamond t.$$

Proof:

Theorem (Mean Value Theorem): Let $f,g \colon \mathbb{T} \rightarrow \mathbb{R}$ be bounded and $\Diamond$-integrable on $[a,b] \cap \mathbb{T}$, and let $g$ be nonnegative or nonpositive on $[a,b] \cap \mathbb{T}$. Let $m$ and $M$ be the infimum and supremum respectively of $f$. Then there exists a real number $K$ satisfying $m \leq K \leq M$ such that $$\displaystyle\int_a^b (fg)(t) \Diamond t = K \displaystyle\int_a^b g(t) \Diamond t.$$

Proof: