Combining two probability distributions I have a variable $X$. In a measurement $A$, $X$ follows the normal distribution $N_1$ with mean $m_1$ and standard deviation $\sigma_1$. In a similar measurement $B$, $X$ follows another normal distribution $N_2$ with mean $m_2$ and standard deviation $\sigma_2$. 
In this case, what will be the combined or joint probability distribution of $X$? Will it be $N_1+N_2$  or $N_1 N_2$ ? 

(Addition) Let's assume $A$ and $B$ are independent measurements. We can think about a situation when a measurer $A$ comes and measure the distribution of $X$ and then next a person $B$ comes and measures the distribution again. Both measurers measure independently. 
The question is what will be the true probability distribution of $X$ in this case? We assume that the measurement of $A$ and $B$ are equally reliable. 

(Paraphrasing a comment from r.e.s.) If person C receives reports from equally-reliable observers A and B, stating their respective independent judgements about X (in the form of the stated normal distributions), then how does C combine these reports to form a fair and unbiased judgement about X?
The work by Clemen &amp; Winkler is not for this situation.  For example, if $P=(1,0,0,0)$ is a probability distribution over a $4$-element set, and $Q=(0.5, 0.3, 0.2, 0)$ is another independently obtained probability distribution over that set, then the probability distribution, $F(P,Q)$, resulting from combining information in $P$ and $Q$, should be $(1,0,0,0)$ because $P$ already has conclusive information on the set elements, that cannot be further "improved" by another observation.  In other words, any $0$-value occurring in $P$ (or $Q$) must result in a $0$-value in $F(P,Q)$ at the same position (the same for any $1$-value logically follows from this).  Also, the identity of $F$ should be the uniform distribution $(0.25, 0.25, 0.25, 0.25)$, as that is the most inconclusive distribution.  Aggregating $P$ and $Q$ by taking their weighted arithmetic or geometric mean does not achieve this, as in most works like Clemen &amp; Winkler.  Could someone please suggest how such a function $F$ should be defined?  We must assume that $P$ and $Q$ are consistent, i.e. if one of them has a $0$-value in a certain position, then the other does not have a $1$-value in the same position, because our two observations cannot contradict each other.  I am not an expert in probability theory, so please pardon any inaccurate terminology usage.  I would appreciate any help with finding such an $F$.  Thanks!
This is the standard problem of combining two independent pieces of evidence via Dempster's rule of combination in belief functions. Please refer to the theory of linear belief function in Wikipedia for the formula of combining two normal distributions.