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In the context of Signal Processing White Noise is usually defined using a single intuitive property: A random process with constant magnitude, for any frequency, of its power spectrum:

If $ w \left( t \right) $ is a white noise then its power spectrum is given by $ {S}_{w} \left( f \right) = {\sigma}^{2} $. In communication it is common to use $ {S}_{w} \left( f \right) = \frac{ {N}_{0} }{2} $.

Using Wiener Khinchin Theorem one would conclude that the Auto Correlation of the process, which is assumed to be a Stationary Process in the wide sense is given by $ {R}_{ww} \left( \tau \right) = {\sigma}^{2} \delta \left( t \right) $.

In the context of Signal Processing we only need the model of Continuous White Noise in order to analyze the output of Linear System with finite frequency support. For this context, the above model is enough.

Yet, if one wants to be rigorous, mathematically, the above model is not enough.
For instance, if one wants the Wiener Khinchin Theorem to hold one must have, arbitrarily small, correlation in the auto correlation function.
So the move from the power density to the auto correlation, as done above, is not justified Mathematically (See for instance "Roy M. Howard - On Defining White Noise").
So the beast, called White Noise, is not plausible both Physically (Which is OK, we are dealing with Math), yet it is not coherent Mathematically, it requires more delicate work. Yet, again, in the context of Signal Processing, this is enough as only deal with it in the context of going through a Linear System with limited bandwidth support.

A more rigorous derivation can be done by defining the (Gaussian) White Noise as the derivative of Wiener Process.
Wiener Process has some properties which makes it interesting:

1. The difference of 2 samples is Gaussian.
2. The samples / realization path is continuous (In the almost surely sense).

Then we can define Gaussian White noise as the derivative of the Wiener Process. This will yield a more coherent definition (For instance, it indeed defines a distribution).

In the context of Signal Processing White Noise is usually defined using a single intuitive property: A random process with constant magnitude, for any frequency, of its power spectrum:

If $ w \left( t \right) $ is a white noise then its power spectrum is given by $ {S}_{w} \left( f \right) = {\sigma}^{2} $. In communication it is common to use $ {S}_{w} \left( f \right) = \frac{ {N}_{0} }{2} $.

Using Wiener Khinchin Theorem one would conclude that the Auto Correlation of the process, which is assumed to be a Stationary Process in the wide sense is given by $ {R}_{ww} \left( \tau \right) = {\sigma}^{2} \delta \left( t \right) $.

In the context of Signal Processing we only need the model of Continuous White Noise in order to analyze the output of Linear System with finite frequency support. For this context, the above model is enough.

Yet, if one wants to be rigorous, mathematically, the above model is not enough.
For instance, if one wants the Wiener Khinchin Theorem to hold one must have, arbitrarily small, correlation in the auto correlation function.
So the move from the power density to the auto correlation, as done above, is not justified Mathematically (See for instance "Roy M. Howard - On Defining White Noise").
So the beast, called White Noise, is not plausible both Physically (Which is OK, we are dealing with Math), yet it is not coherent Mathematically, it requires more delicate work. Yet, again, in the context of Signal Processing, this is enough as only deal with it in the context of going through a Linear System with limited bandwidth support.

A more rigorous derivation can be done by defining the (Gaussian) White Noise as the derivative of Wiener Process.
Wiener Process has some properties which makes it interesting:

1. The difference of 2 samples is Gaussian.
2. The samples / realization path is continuous (In the almost surely sense).

Then we can define Gaussian White noise as the derivative of the Wiener Process. This will yield a more coherent definition (For instance, it indeed defines a distribution).

Remark I'd be happy if those who -1 will specify why.
I will try to explain the answer.

The answer has the following logic:

• We want a random process. Specifically a stationary random process (Even if only in the wide sense, there are subtleties to that, but let's skip this).
• It should obey Wiener Khinchin Theorem we need to be able to analyze it using spectral methods.

A reasonable way to do so is define a random process as following.

• Let $ v \left( t \right) $ be a stationary random process.
• For any pair $ {t}_{i} \neq {t}_{j} $ we would like to have $ \mathbb{E} \left[ v \left( {t}_{i} \right) v \left( {t}_{j} \right) \right] = 0 $.

If we can have that, great, we have a White Noise.
It turns out we can't! Why? Read the reference.
It has to do with convergence of a limit and order of integration and the limit operator.

So, what can have?
We can define $ \epsilon > 0 $ arbitrary small for which the auto correlation function is not zero.
It will give us a support, arbitrary large, in the PSD with constant value.
This model of White Noise will obey the needs of Signal Processing, namely for any Linear System with finite bandwidth the properties will be as we know them yet indeed our process will obey the properties of a valid random process, stationarity included.

In the context of Signal Processing White Noise is usually defined using a single intuitive property: A random process with constant magnitude, for any frequency, of its power spectrum:

If $ w \left( t \right) $ is a white noise then its power spectrum is given by $ {S}_{w} \left( f \right) = {\sigma}^{2} $. In communication it is common to use $ {S}_{w} \left( f \right) = \frac{ {N}_{0} }{2} $.

Using Wiener Khinchin Theorem one would conclude that the Auto Correlation of the process, which is assumed to be a Stationary Process in the wide sense is given by $ {R}_{ww} \left( \tau \right) = {\sigma}^{2} \delta \left( t \right) $.

In the context of Signal Processing we only need the model of Continuous White Noise in order to analyze the output of Linear System with finite frequency support. For this context, the above model is enough.

Yet, if one wants to be rigorous, mathematically, the above model is not enough.
For instance, if one wants the Wiener Khinchin Theorem to hold one must have, arbitrarily small, correlation in the auto correlation function.
So the move from the power density to the auto correlation, as done above, is not justified Mathematically (See for instance "Roy M. Howard - On Defining White Noise").
So the beast, called White Noise, is not plausible both Physically (Which is OK, we are dealing with Math), yet it is not coherent Mathematically, it requires more delicate work. Yet, again, in the context of Signal Processing, this is enough as only deal with it in the context of going through a Linear System with limited bandwidth support.

A more rigorous derivation can be done by defining the White Noise as the derivative of Wiener Process.
Wiener Process has some properties which makes it interesting:

1. The difference of 2 samples is Gaussian.
2. The samples / realization path is continuous (In the almost surely sense).

Then we can define Gaussian White noise as the derivative of the Wiener Process. This will yield a more coherent definition (For instance, it indeed defines a distribution).

In the context of Signal Processing White Noise is usually defined using a single intuitive property: A random process with constant magnitude, for any frequency, of its power spectrum:

If $ w \left( t \right) $ is a white noise then its power spectrum is given by $ {S}_{w} \left( f \right) = {\sigma}^{2} $. In communication it is common to use $ {S}_{w} \left( f \right) = \frac{ {N}_{0} }{2} $.

Using Wiener Khinchin Theorem one would conclude that the Auto Correlation of the process, which is assumed to be a Stationary Process in the wide sense is given by $ {R}_{ww} \left( \tau \right) = {\sigma}^{2} \delta \left( t \right) $.

In the context of Signal Processing we only need the model of Continuous White Noise in order to analyze the output of Linear System with finite frequency support. For this context, the above model is enough.

Yet, if one wants to be rigorous, mathematically, the above model is not enough.
For instance, if one wants the Wiener Khinchin Theorem to hold one must have, arbitrarily small, correlation in the auto correlation function.
So the move from the power density to the auto correlation, as done above, is not justified Mathematically (See for instance "Roy M. Howard - On Defining White Noise").
So the beast, called White Noise, is not plausible both Physically (Which is OK, we are dealing with Math), yet it is not coherent Mathematically, it requires more delicate work. Yet, again, in the context of Signal Processing, this is enough as only deal with it in the context of going through a Linear System with limited bandwidth support.

A more rigorous derivation can be done by defining the (Gaussian) White Noise as the derivative of Wiener Process.
Wiener Process has some properties which makes it interesting:

1. The difference of 2 samples is Gaussian.
2. The samples / realization path is continuous (In the almost surely sense).

Then we can define Gaussian White noise as the derivative of the Wiener Process. This will yield a more coherent definition (For instance, it indeed defines a distribution).

In the context of Signal Processing White Noise is usually defined using a single intuitive property: A random process with constant magnitude, for any frequency, of its power spectrum:

If $ w \left( t \right) $ is a white noise then its power spectrum is given by $ {S}_{w} \left( f \right) = {\sigma}^{2} $. In communication it is common to use $ {S}_{w} \left( f \right) = \frac{ {N}_{0} }{2} $.

Using Wiener Khinchin Theorem one would conclude that the Auto Correlation of the process, which is assumed to be a Stationary Process in the wide sense is given by $ {R}_{ww} \left( \tau \right) = {\sigma}^{2} \delta \left( t \right) $.

In the context of Signal Processing we only need the model of Continuous White Noise in order to analyze the output of Linear System with finite frequency support. For this context, the above model is enough.

Yet, if one wants to be rigorous, mathematically, the above model is not enough.
For instance, if one wants the Wiener Khinchin Theorem to hold one must have, arbitrarily small, correlation in the auto correlation function.
So the move from the power density to the auto correlation, as done above, is not justified Mathematically (See for instance "Roy M. Howard - On Defining White Noise").
So the beast, called White Noise, is not plausible both Physically (Which is OK, we are dealing with Math), yet it is not coherent Mathematically, it requires more delicate work. Yet, again, in the context of Signal Processing, this is enough as only deal with it in the context of going through a Linear System with limited bandwidth support.

A more rigorous derivation can be done by defining the White Noise as the derivative of Wiener Process.
Wiener Process has some properties which makes it interesting:

1. The difference of 2 samples is Gaussian.
2. The samples / realization path is continuous (In the almost surely sense).

Then we can define Gaussian White noise as the derivative of the Wiener Process. This will yield a more coherent definition (For instance, it indeed defines a distribution).

In the context of Signal Processing White Noise is usually defined using a single intuitive property: A random process with constant magnitude, for any frequency, of its power spectrum:

If $ w \left( t \right) $ is a white noise then its power spectrum is given by $ {S}_{w} \left( f \right) = {\sigma}^{2} $. In communication it is common to use $ {S}_{w} \left( f \right) = \frac{ {N}_{0} }{2} $.

Using Wiener Khinchin Theorem one would conclude that the Auto Correlation of the process, which is assumed to be a Stationary Process in the wide sense is given by $ {R}_{ww} \left( \tau \right) = {\sigma}^{2} \delta \left( t \right) $.

In the context of Signal Processing we only need the model of Continuous White Noise in order to analyze the output of Linear System with finite frequency support. For this context, the above model is enough.

Yet, if one wants to be rigorous, mathematically, the above model is not enough.
For instance, if one wants the Wiener Khinchin Theorem to hold one must have, arbitrarily small, correlation in the auto correlation function.
So the move from the power density to the auto correlation, as done above, is not justified Mathematically (See for instance "Roy M. Howard - On Defining White Noise").
So the beast, called White Noise, is not plausible both Physically (Which is OK, we are dealing with Math), yet it is not coherent Mathematically, it requires more delicate work. Yet, again, in the context of Signal Processing, this is enough as only deal with it in the context of going through a Linear System with limited bandwidth support.

A more rigorous derivation can be done by defining the White Noise as the derivative of Wiener Process.
Wiener Process has some properties which makes it interesting:

1. The difference of 2 samples is Gaussian.
2. The samples / realization path is continuous (In the almost surely sense).

Then we can define Gaussian White noise as the derivative of the Wiener Process. This will yield a more coherent definition (For instance, it indeed defines a distribution).