# Diamond integral

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:** █