Claim that a distribution is almost Gaussian I prove a some theorem under the assumption that some random variable X is Gaussian. Now in practice, in my experiments section I have real-world samples from X, and I want to claim that indeed these samples form approximately a Gaussian distribution, therefore the conditions of the theorem approximately hold. Of course in real life nothing is exactly a Gaussian, only approximately. 
Now there are some normality tests such as [link], but they all have the null hypothesis that the samples were drawn from a Gaussian distribution, therefore I can at most fail to reject this hypothesis (when p>0.05), but it still doesn't mean that my samples were drawn from an almost Gaussian distribution. 
Any ideas what is the standard way to quantitatively show that my samples were drawn from an approximately Gaussian distribution?
 A: A normal QQ plot is always a good idea in view of the problematic nature of normality tests. The QQ plot compares sample quantiles with theoretical quantiles and so if your data lie on a straight line or approximately a straight line then you can claim approximate normality. See also How to interpret a QQ plot.
A: This is not a full answer to the question, because it denies the premise of the question.
Some things can be very close to Gaussian without basic properties that follow from normality holding. For example, consider a mixture that's $(1-\epsilon)$ from a standard normal and $\epsilon$ from a standard Cauchy, for arbitrarily small but fixed $\epsilon>0$. 
Then (for example) our distribution has no moments. So while it may often be the case that for a wide class of distribution functions more or less similar to the normal some property may hold approximately, you may be able to get arbitrarily close (in a particular sense) and not have even simple properties that hold for the normal.


Some properties will hold just fine in a situation like this, but others will not.
This suggests that you have to be very careful about how sensitive the properties are to the particular kinds of non-normality you might have.
