Delta integral

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There are a few equivalent definitions of $\Delta$-integration.

Cauchy $\Delta$-integral

Let $\mathbb{T}$ be a time scale. We say that $f$ is regulated if its right-sided limits exist (i.e. are finite) at all right-dense points of $\mathbb{T}$ and its left-sided limits exist (i.e. are finite) at all left-dense points of $\mathbb{T}$. We say that $f$ is pre-differentiable with region of differentiation $D$ if $D \subset \mathbb{T}^{\kappa}$, $\mathbb{T}^{\kappa} \setminus D$ is countable with no right-scattered elements of $\mathbb{T}$, and $f$ is $\Delta$-differentiable at each $t \in D$. Now suppose that $f$ is regulated. It is known that there exists a function $F$ which is pre-differentiable with region of differentiation $D$ such that $F^{\Delta}(t)=f(t)$. We define the indefinite integral of a regulated function $f$ by $$\displaystyle\int f(t) \Delta t = F(t)+C$$ for an arbitrary constant $C$.

Now we define the definite integral, i.e. the Cauchy integral, by the formula $$\displaystyle\int_s^t f(\tau) \Delta \tau = F(t)-F(s)$$ for all $s,t \in \mathbb{T}$.

A function $F \colon \mathbb{T}\rightarrow \mathbb{R}$ is called an antiderivative of $f \colon \mathbb{T}\rightarrow \mathbb{R}$ if $F^{\Delta}(t)=f(t)$ for all $t \in \mathbb{T}^{\kappa}$. It is known that all rd-continuous functions possess an antiderivative, in particular if $t_0 \in \mathbb{T}$ then $F$ defined by $$F(t) = \displaystyle\int_{t_0}^t f(\tau) \Delta \tau$$ is an antiderivative of $f$.

Riemann $\Delta$-integral

Lebesgue $\Delta$-integral

Properties of $\Delta$-integrals

Delta integral from t to sigma(t)
Delta integral is linear
Interchanging limits of delta integral
Delta integrals are additive over intervals

Theorem (Integration by Parts,I): The following formula holds: $$\int_a^b f(\sigma(t))g^{\Delta}(t) \Delta t = (fg)(b) - (fg)(a) - \int_a^b f^{\Delta}(t)g(t) \Delta t.$$


Theorem (Integration by Parts,II): The following formula holds: $$\int_a^b f(t) g^{\Delta}(t) \Delta t = (fg)(b) - (fg)(a) - \int_a^b f^{\Delta}(t) g(\sigma(t)) \Delta t.$$


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


Theorem: If $|f(t)| \leq g(t)$ on $[a,b)$ then $$\left| \int_a^b f(t) \Delta t \right| \leq \int_a^b g(t) \Delta t$$


Theorem: If $f(t) \geq 0$ for all $a \leq t < b$ then $$\displaystyle\int_a^b f(t) \Delta t \geq 0.$$


Theorem (Fundamental theorem of calculus,I): The following formula holds: $$\int_a^b f^{\Delta}(t) \Delta t = f(b)-f(a).$$


Theorem (Fundamental theorem of calculus,II): The following formula holds: $$\left( \int_{t_0}^x f(\tau) \Delta \tau) \right)^{\Delta} = f(x).$$