There are some authors that define *idealized 1-dimensional white noise* as a stochastic process $\{\dot W_t\}_{t\geq 0}$ as a process with the following properties:

 1. It is a mean zero Gaussian process.
 2. If $t_1\neq t_2$ then $\dot W_{t_1}$ and $\dot W_{t_2}$ are independent.
 3. It is a stationary process. 

However, in several references I have read so far, I do not see an explicit construction of this process with index set $T=\mathbb{R}^+$. Rather, white noise is defined in the following two ways.  

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***White noise indexed by a $\sigma$-algebra of a $\sigma$-finite measure space.*** 

Let $(M,\mathcal{F},\sigma)$ be a $\sigma$-finite measure space and let $T$ be the collection of all sets in $\mathcal{F}$ of finite measure. White noise $\{W_A\}_{A\in T}$ is then defined as mean zero Gaussian process such that the following hold. 
 
 - For each $A\in T$, $W_A$ is a normal random variable with zero mean and variance $\sigma(A)$. 
 - If $A, B\in T$ and $A\cap B=\emptyset$ then $W_A$ and $W_B$ are independent.
 - For all $A,B\in T$ we have that $W(A\cup B)=W(A)+W(B)-W(A\cap B)$.

Some version of the Bochner-Minlos theorem to get the existence of such a process. It has the property that for any $A,B\in T$ that $\text{Cov}(W_A,W_B)=\sigma(A\cap B)$. Note that if $M=\mathbb{R}$, $\mathcal{F}=\mathcal{B}$ is the Borel $\sigma$-algebra and $\sigma=\lambda$ is the Lebesgue measure then the process 
$$\{B_t\}_{t\geq 0}:=\left\{W\left((0,t]\right)\right\}_{t\geq 0}$$ is just regular Brownian motion. 

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***White noise indexed by Shwartz functions.*** 

Let $T:=\mathcal{S}(\mathbb{R})$ be the Shwartz space rapidly decreasing smooth functions.
Let $\Omega:=\mathcal{S'}(\mathbb{R})$ be the space of tempered distributions equipped with the cylinder algebra $\mathscr{C}$. A version of Bochner-Minlos theorem then says that there is a probability measure $\mathbb{P}$ on $(\Omega,\mathscr{C})$ such that 
$$\int_\Omega\langle \omega,\varphi\rangle d\mathbb{P}(\omega)=\exp\left(-\frac{1}{2}\|\varphi\|_{L^2(\mathbb{R})}\right),\;\;\;\text{ for all }\varphi\in T.$$ 
White noise is then defined as 
$$W(t,\omega):=\langle \omega, \varphi \rangle,\;\;\;\text{ for all }(t,\omega)\in T\times \Omega.  $$
Note that for $\omega\in\Omega$ and for any $\varphi\in L^2(\mathbb{R})$ one can define $\langle \omega,\varphi\rangle$ by using a sequence of functions $\{\varphi_n\}$ in $T$ that converge to $\varphi$. This allows to defined the following:
$$B_t(\omega)=\langle \omega,\mathbf{1}_{(0,t]}\rangle,\;\;\;\text{ for all }t\in\mathbb{R}^+. $$
Then $\{B_t\}$ is just one dimensional Brownian motion.

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***The Question:***

As you can see, one can use white noise (defined in any of the above ways) to construct Brownian motion. 

 - Can we use white noise $W$ to construct idealized white noise $\dot W$? 
 - Is there another way to construct $\dot W$?


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***My idea:*** (can skip this part)

We can use the second construction of white noise above to define a process called the  *smoothed white noise process*. It is defined as follows: For $\varphi\in L^2(\mathbb{R})$ and any $t\in \mathbb{R}$ let $$\varphi_t(x)=\varphi(t-x),\;\;\;\text{ for }x\in \mathbb{R}.$$ Then define 
$$W_\varphi(t,\omega)=W(\varphi_t,\omega)=\langle \omega,\varphi_t\rangle,\;\;\;\text{ for all }(t,\omega)\in T\times\Omega. $$

$W_\varphi$ is then a zero mean stationary Gaussian process with constant variance equal to $\|\varphi\|_{L^2}$. It has the properties that for any $t_1\neq t_2$, if $\varphi_{t_1}$ and $\varphi_{t_2}$ have disjoint supports then $W(t_1,\cdot)$ and $W(t_2,\cdot)$ are independent. 

So I was thinking, if we take a sequence $\{\varphi_n\}$ in $C_c^\infty(\mathbb{R})$ with shrinking supports, then would the sequence of smoothed white noises $\{W_{\varphi_n}\}$ converge (in a appropriate sense) to an idealized white noise $\dot W$?