The Romans Saw Jerusalem as Part of Judea, that is as Jewish

Pliny the Elder was one of the greatest scholars of ancient Rome. His most famous work is the Natural History [Historia Naturalis. He wrote about Jerusalem in that book. Here is one many mentions of Jerusalem in it:

Jerusalem, by far the most famous city of the East, not merely of Judea.

Hierosolyma longe clarissima urbium Orientis, non Ivdaeae modo.
[Historia Naturalis, V:xv:70]

Here is another quote which refers to Machaerus [מכור], a Jewish mountain fortress east of the Dead Sea. Yes, Machaerus is east of the Dead Sea:

Machaerus, at one time next to Jerusalem the most important fortress in Judea.

Machaerus, secunda quondam arx Ivdaeae ab Hierosolymis.
[Historia Naturalis, V:xv:72]

Note the respect that Pliny the Elder, Gaius Plinius Secundus [23-79 CE], showed for Jerusalem and for Judea, and thus for the Jews. European attitudes toward the Jews have changed for the worse since then. Also note that in Pliny's time, the Romans called the country where Jerusalem was located Judea. It was not until the Roman defeat of the Jewish Bar Kokhba Revolt, about  56 years after the death of Pliny the Elder, that Emperor Hadrian changed the name of the Provincia Ivdaea to Provincia Syria Palaestina. a name that evolved over the centuries into simply Palaestina. Bear in mind that Hadrian imposed the new name on the country as a punishment for its people, the Jews, who had bravely rebelled against the Roman Empire. Use of the name today is a kind of taking sides after the fact with the Roman imperialists. Hadrian likewise changed the name of Jerusalem to Aelia Capitolina, bringing together his own clan or gens name, Aelius, with the name of the famous hill in Rome where the Senate was located [now the Campidoglio].
Another point, Judea was the Roman name for the land that the Jews called the Land of Israel.  A Jew was called Ivdaeus and Jews were Ivdaei.

It is also significant that in its original form, Syria Palaestina meant Palestinian Syria, with the word Palaestina being an adjective, whereas Judea was seen as part of Syria which was an old geographical conception of Greeks and Romans. As indicated, this changed over the centuries, with Palaestina --coming to be seen as a noun rather than as an adjective-- as the Roman name for the Land of Israel, which the Romans had earlier had called Judea [Ivdaea], the country's name in the heyday of the Empire. Judea stretched along both sides of the Jordan River --as we see from Machaerus [מכור] being east of the Dead Sea, a southern extension of the Jordan. Judea to be sure stretched much further west of the Jordan than east of the River. And today Israel only has its ancient lands that were to the west of the River.

This matter of ancient names of the country and the country in which Jerusalem was located is of course of great importance nowadays when Mahmoud Abbas, potentate of the Palestinian Authority, utilizes a false history of this country in order to deny Jewish rights in Jerusalem, in any part of Jerusalem. Yes, he denies Jewish rights in any part of the Holy City. Otherwise why would he object to President Trump recognizing Jerusalem as Israel's capital, which it has been since 1948, and whereas Trump is transferring the embassy for the time being to the present main consular building of the  USA consulate general in the city which is located in a place that Israel has held since 1948 and was not acquired by Israel in the 1967 Six Day War?
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For those who know the New Testament, the country is called Land of Israel in Matthew I:20-22. Elsewhere in the NT the country is called Judea, following the Roman usage. In some places, in the NT, Judea is used together with Samaria, following the narrower Jewish usage of the place name Judea which saw Judea as the continuation of the name Judah for the southern kingdom of the Israelites/Jews.

See a Roman document, an truly ancient document fairly well preserved for its age, that shows that ancient Rome called this country Judea or IVDAEA [here].

About the name Aelia Capitolina:
Aelia was still in use when the Arabs conquered the country. The Arabs retained pre-Arab place names in the conquered territories. Hence, they called Jerusalem at first Iliya, their pronunciation of Aelia (as proven by their coins, inter alia). Only a few hundred years after the conquest did they begin to use al-Quds and Bayt al-Maqdis which were copied from Jewish terms, haQodesh and Beyt haMiqdash, which originally referred to holiness and the Temple, and later were applied by Jews to the city of Jerusalem.