Which law in probability theory states the following? Which law in probability theory states the following?
If we have a large enough number of samples, their histogram function converges  their true probability density function. (for a continuous random variable)
I know that "In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed."
But this law is just about expected value. It does not include variance or probability density function.
 A: I worry that the statement you've made doesn't quite make sense. Exactly what is the definition of "the probability density function of the samples"? For instance, if you take the uniform probability on the unit interval, after $k$ samples, you'll might say that that "prob. density of the samples" is a function that's $1/k$ at each of the $k$ sample points...but such a function won't converge to the everywhere-one function, for the limit of these individual functions can be nonzero on at most a countable number of points. 
As for your followup question about standard deviation, I believe that the answer is "no," for there are distributions whose standard deviation is infinite, but the sample-standard-deviation will always be finite, hence not "close" to the SD of the underlying distribution. 
A: Your statement is rather imprecise. One could assume the following (but perhaps this is restrictive) : the density $f(x)$ has support on $[a,b]$. Let $g_{n,m}(x)$ be the histogram constructed by taking $n$ samples and dividing the support interval in $m$ segments of same length $h=(b-a)/m$. Let $x_i$ be the center point of each histogram segment.
Then $g_{n,m}(x_i)$ is a Binomial $B(n,p)$ variable with  $p=\int_{x_i-h/2}^{x_i+h/2} f(x) dx = I_{x_i,h}\approx h f(x_i)$
This approximation holds if $h \to 0$ and the function has bounded derivative.
Let $$w_{n,m}(x_i)=\frac{g_{n,m}(x_i)}{nh}$$ be the normalized histogram.
Then $$E\left(w_{n,m}(x_i)\right)= \frac{ I_{x_i,h}}{h}\approx f(x_i)$$
$$Var\left(w_{n,m}(x_i)\right)= \frac{1}{n h^2} I_{x_i,h}(1-I_{x_i,h})\approx \frac{ f(x_i)}{n h}$$
Then, if the above condition holds, and $h\to 0$, and $n h \to \infty$, the histogram is asymptotically unbiased, and it's variance tends to zero, hence  it converges in mean square (and hence in probability).
A: Probably the result closest to what you're saying would the Glivenko-Cantelli theorem: https://en.wikipedia.org/wiki/Glivenko%E2%80%93Cantelli_theorem.  This states that the empirical distribution function of a random sample converges in a certain sense to the true distribution function as the sample size tends to infinity.
