Why the property $(b)$ in Preposition 14 in this paper on surreal integration is considered essential?

The Preposition lists the desired properties of the surreal integration, and among others lists the following:

$$\int_a^b (\alpha f+\beta g)=\alpha \int_a^b f + \beta \int_a^b g,$$

where $\alpha, \beta \in \text{No}$.

While I have no objections regarding integral's distribution against addition, the idea of moving an infinite factor out from under the integral unchanged seems completely unjustified to me.

Let's look at a simple example. Assume the canonical embedding of Hardy field into surreals, so that the germ of the identity function $f(x)=x$ at infinity corresponds to $\omega$.

Let $u(x)={\begin{cases}1,&{\text{if }}x=0\\ 0,&{\text{if }}x\ne 0\end{cases}}$ to be the unit-impulse function.

Let us consider the integral $\int_{-1}^1\omega u(x)dx$.

Since we have embedded germs into surreals, $\omega=\int_0^\infty 1 dx$. Following Fourier (or Laplace) transform,

$$\int_{-1}^1\omega u(x)dx=\int_{-1}^1 \pi\delta(x) dx=\pi\ne \omega\int_{-1}^1 u(x)dx=0$$

So, the property of moving an infinite factor from under the integral naturally does not hold.

If so, why does the linked paper and other papers on surreal integration insist on it?

Why the property $(b)$ in Proposition 14 in this paper on surreal integration is considered essential?

The Proposition lists the desired properties of the surreal integration, and among others lists the following:

$$\int_a^b (\alpha f+\beta g)=\alpha \int_a^b f + \beta \int_a^b g,$$

where $\alpha, \beta \in \text{No}$.

While I have no objections regarding integral's distribution against addition, the idea of moving an infinite factor out from under the integral unchanged seems completely unjustified to me.

Let's look at a simple example. Assume the canonical embedding of Hardy field into surreals, so that the germ of the identity function $f(x)=x$ at infinity corresponds to $\omega$.

Let $u(x)={\begin{cases}1,&{\text{if }}x=0\\ 0,&{\text{if }}x\ne 0\end{cases}}$ to be the unit-impulse function.

Let us consider the integral $\int_{-1}^1\omega u(x)dx$.

Since we have embedded germs into surreals, $\omega=\int_0^\infty 1 dx$. Following Fourier (or Laplace) transform,

$$\int_{-1}^1\omega u(x)dx=\int_{-1}^1 \pi\delta(x) dx=\pi\ne \omega\int_{-1}^1 u(x)dx=0$$

So, the property of moving an infinite factor from under the integral naturally does not hold.

If so, why does the linked paper and other papers on surreal integration insist on it?

Why the property $(b)$ in Preposition 14 that can be found in this paper on surreal integration is considered essential?

The Preposition lists the desired properties of the surreal integration, and among others lists the following:

$$\int_a^b (\alpha f+\beta g)=\alpha \int_a^b f + \beta \int_a^b g,$$

where $\alpha, \beta \in \text{No}$.

While I have no objections regarding integral's distribution against addition, the idea of moving an infinite factor out from under the integral unchanged seems completely unjustified to me.

Let's look at a simple example. Assume the canonical embedding of Hardy field into surreals, so that the germ of the identity function $f(x)=x$ at infinity corresponds to $\omega$.

Let $u(x)={\begin{cases}1,&{\text{if }}x=0\\ 0,&{\text{if }}x\ne 0\end{cases}}$ to be the unit-impulse function.

Let us consider the integral $\int_{-1}^1\omega u(x)dx$.

Since we have embedded germs into surreals, $\omega=\int_0^\infty 1 dx$. Following Fourier (or Laplace) transform,

$$\int_{-1}^1\omega u(x)dx=\int_{-1}^1 \pi\delta(x) dx=\pi\ne \omega\int_{-1}^1 u(x)dx=0$$

So, the property of moving an infinite factor from under the integral naturally does not hold.

If so, why does the linked paper and other papers on surreal integration insist on it?

Why the property $(b)$ in Preposition 14 in this paper on surreal integration is considered essential?

The Preposition lists the desired properties of the surreal integration, and among others lists the following:

$$\int_a^b (\alpha f+\beta g)=\alpha \int_a^b f + \beta \int_a^b g,$$

where $\alpha, \beta \in \text{No}$.

While I have no objections regarding integral's distribution against addition, the idea of moving an infinite factor out from under the integral unchanged seems completely unjustified to me.

Let's look at a simple example. Assume the canonical embedding of Hardy field into surreals, so that the germ of the identity function $f(x)=x$ at infinity corresponds to $\omega$.

Let $u(x)={\begin{cases}1,&{\text{if }}x=0\\ 0,&{\text{if }}x\ne 0\end{cases}}$ to be the unit-impulse function.

Let us consider the integral $\int_{-1}^1\omega u(x)dx$.

Since we have embedded germs into surreals, $\omega=\int_0^\infty 1 dx$. Following Fourier (or Laplace) transform,

$$\int_{-1}^1\omega u(x)dx=\int_{-1}^1 \pi\delta(x) dx=\pi\ne \omega\int_{-1}^1 u(x)dx=0$$

So, the property of moving an infinite factor from under the integral naturally does not hold.

If so, why does the linked paper and other papers on surreal integration insist on it?

Why the property $(b)$ in Preposition 14 that can be found in this paper on surreal integration is considered essential?

The Preposition lists the desired properties of the surreal integration, and among others lists the following:

$$\int_a^b (\alpha f+\beta g)=\alpha \int_a^b f + \beta \int_a^b g,$$

where $\alpha, \beta \in \text{No}$.

While I have no objections regarding integral's distribution against addition, the idea of moving an infinite factor out from under the integral unchanged seems completely unjustified to me.

Let's look at a simple example. Assume the canonical embedding of Hardy field into surreals, so that the germ of the identity function $f(x)=x$ at infinity corresponds to $\omega$.

Let $u(x)={\begin{cases}1,&{\text{if }}x=0\\ 0,&{\text{if }}x\ne 0\end{cases}}$ to be the unit-impulse function.

Let us consider the integral $\int_{-1}^1\omega u(x)dx$.

Since we have embedded germs into surreals, $\omega=\int_0^\infty 1 dx$. Following Fourier transform,

$$\int_{-1}^1\omega u(x)dx=\int_{-1}^1 \pi\delta(x) dx=\pi\ne \omega\int_{-1}^1 u(x)dx=0$$

So, the property of moving an infinite factor from under the integral naturally does not hold.

If so, why does the linked paper and other papers on surreal integration insist on it?

Why the property $(b)$ in Preposition 14 that can be found in this paper on surreal integration is considered essential?

The Preposition lists the desired properties of the surreal integration, and among others lists the following:

$$\int_a^b (\alpha f+\beta g)=\alpha \int_a^b f + \beta \int_a^b g,$$

where $\alpha, \beta \in \text{No}$.

While I have no objections regarding integral's distribution against addition, the idea of moving an infinite factor out from under the integral unchanged seems completely unjustified to me.

Let's look at a simple example. Assume the canonical embedding of Hardy field into surreals, so that the germ of the identity function $f(x)=x$ at infinity corresponds to $\omega$.

Let $u(x)={\begin{cases}1,&{\text{if }}x=0\\ 0,&{\text{if }}x\ne 0\end{cases}}$ to be the unit-impulse function.

Let us consider the integral $\int_{-1}^1\omega u(x)dx$.

Since we have embedded germs into surreals, $\omega=\int_0^\infty 1 dx$. Following Fourier (or Laplace) transform,

$$\int_{-1}^1\omega u(x)dx=\int_{-1}^1 \pi\delta(x) dx=\pi\ne \omega\int_{-1}^1 u(x)dx=0$$

So, the property of moving an infinite factor from under the integral naturally does not hold.

If so, why does the linked paper and other papers on surreal integration insist on it?