Mettā, karuṇā, muditā and upekkhā are called the four brahmavihārā, often translated as divine abodes or abidings, sublime attitudes, abodes of Brahma.
Brahma is according to the texts the name of a higher god and vihāra means dwelling, as we are already used seeing viharati in 'On one occasion the exalted one dwells', so we can call it 'dwellings of Brahma'. If one acts like Brahma, have a mindset like Brahma, then one might end up in such situation. MN 96 says this goes for anyone. So both commoners and trainees can end up in heavens. The principles for these are seen as godly, divine, saintly, holy and are thus higher than what people usually would cultivate or aim for (see also "Ariyasāvaka").
While brahmavihāra is hardly ever used in the texts SN 54.11 shows that the concentration from meditation on the in plus out breathe is also a brahmavihāra. Thus the brahmavihāra are not the four, but the four are brahmavihārā. It is a dwelling from a higher principle what makes a dwelling Brahma's dwelling (see also "Bhava"). But as just these four are grouped, not another number nor another combination, there must be a reason to do so. Let's go into this.
Mettā
Mettā is often found translated as loving-kindness, benevolence, goodwill. The dictionary links it with love. From the texts like "From Another Tradition (AN 3.69)" we know mettā goes against hate. And love and hate are a famous pair. But love doesn't quite fit as it is a bit ambiguous. Perhaps then kindness and the lot got being used but they too don't quite fit. The Pāḷi might have different words for what we see here as ambiguous and thus this might not be the case there. This then gives us some playing room. So, back to love. Love differs from loving, which differs from to be in love, which differs from lover, and so there is this range with love which may or may not quite fit. But if I say 'lovingness' it is already understood differently as it is a quality of mind which is more about indicating care out of love. A mind filled with lovingness? Well, why not? Lovingness fits just fine.
Karuṇā
Karuṇā is often translated as compassion. It is said to mean pity, compassion and sometimes we can find sympathy. MN 62 shows it gives up vihesā which can mean fright. The dictionary gives a whole range for vihesā, including harm, cruelty (perhaps as cause) but for now we don't need to look further into this. However, when the texts talk about compassion the Pāḷi word is anukampa and not karuṇā. And another problem, at least that I have with it, is that pity, compassion, sympathy depend on someone else's misfortune. So they may be good qualities but is it what is meant here? The Brahma quality we could take as having a mind like Brahma, is higher than that of many other gods. So a good quality in itself doesn't mean it is high enough to be considered Brahma's. And such a divine mind, what could it entail? We already have lovingness and we have to come up with three more. But as there is not much to go by let's continue with the other two first to see if this helps sorting things out.
Muditā
Muditā is often translated as empathetic or appreciative joy. The dictionary says it has to do with soft-heartedness, kindliness, sympathy. However, in AN 3.96 we find that muditā is when there is no quarrelling, when the company is in unity, that it is as milk blending with water. And AN 6.113 shows it goes against arati of which the dictionary says it is something like dislike, discontent, aversion. So what is described here is a situation where people seem to be in some sort of agreement, where things blend nicely together, which just describes what harmony is. And to dwell in harmony, wouldn't that be divine?
Upekkhā
Upekkhā is often translated as equanimity. Equanimity can be used to indicate an evenness of mind, balance, calmness. MN 62 shows it gives up friction. It is used describing the third and fourth radiance "Division (SN 45.8)". To me it is about the viewpoint to how things are seen, from objectiveness, rather than that indicating a mental state. Upekkhā is said to come from upa (near, upon, by, above) and ikkha (to see, to look) and meaning "looking from above"; thus from taking a distance. And objectiveness does cover a balanced or impartial aspect. That by stepping back, at a greater distance, things can be looked at more objectively, and more clearly, is something we do understand. First we get to look at the world objectively and so turn away from it, and then from the radiances on we are getting to look objectively at aspects we consider to be of ourselves. To not to take everything to heart, not take everything personal, but to act and look at things from a certain objectiveness, doesn't that already look saintly like?
Back to Karuṇā
So far then we got lovingness, karuṇā, harmony, objectiveness. Does it not seem there is a certain cadence to it? A gradual stilling or a going higher? Then we could say that by stepping back from lovingness we would get to karuṇā; by stepping back from karuṇā, harmony; by stepping back from harmony, objectiveness. And this then could be a reason why they are bounded, as the group of four. As there is not much to go by we now have to make an estimate guess for karuṇā. If possible I would like to stay somewhat in the range of compassion, pity but without the misfortune part as they lean on unwholesome principles which have here already been dealt with. So we are looking for something which has greater distance to lovingness. As we are talking about principles it must be possible to see traces of this in every days' usage too. And we can see people who say they are in love, which involves lovingness, can fall out of it. And if we stay at the good side of things it can be that a certain amiability or friendship remains. Which does indicate a greater distance. And of karuṇā the dictionary indicates to compare it with the Vedic karuṇa nt. (holy) action. So perhaps kindness then, as in acting out of kindness? A mind filled with kindness, wouldn't that be anything deity like?
Mettā, Karuṇā, Muditā, Upekkhā
Lovingness, kindness, harmony, objectiveness. This now flows, right? Yoked to these higher principles, beyond the human norm, where else than beyond the human world could cognition be established. The brahmavihārā are hardly mentioned in the texts but by making use of them we were able to find a translation for karuṇā. The texts do talk about the four apart from them as dwellings of Brahma. As in first getting rid of the unwholesome principles and then promote this goodness (lovingness, kindness, harmony, objectiveness) thus as right exertion. And with right exertion, right meditation, with right meditation, right concentration.
Pāḷi-English
Brahmavihārā dwellings of Brahma.
Mettā (f.) lovingness.
Karuṇā (f.) kindness.
Muditā (f.) harmony.
Upekkhā (f.) objectiveness.