Do you have experience on connecting a Sub-1GHz transceiver to public TV antenna?

It might be good to take a step back. It could be worth mocking up something, like a quick tryout, to see what issues there could be, if you're new to this.

For example, you might find that the performance from a modem, is already as good as it is for your mobile phone. Maybe any bottlenecks are less to do with the antenna.

Once you've gathered some information, then it's best to spend time to design the overall solution first, before getting into this.

What if your solution needs two modems, will you attach a second TV antenna? On second pole on the building? As you can see, just that slight requirement (sure it's hypothetical, I'm using it as an example) would then render the idea impractical now regardless of the technical concerns of your antenna and cable (of which there are for sure concerns; none of that will be ideal unfortunately). By the way, you can buy antennas (either indoor or outdoor) designed for cellular use.

If you're designing all this from scratch, then if it were me, I'd try if possible, as part of the overall design, to locate such things at a suitable position and then run low-cost cable (e.g. cat5, it's less than $1 per meter) all the way to where you want to locate anything else (if you're not say using wireless; I have no idea what your solution is going to look like). There may be reasons not to do that, but it would at least be worth considering (if you have not already ruled it out), since there's a chance it would be more practical. But definitely worth a quick experiment first I think; because I think you're unsure what will work at all, if you're currently considering TV antennas.

Thanks for your valuable opinion.

I have solved the LTE CAT M1 modem problem by simply orienting its location around my home.  Signal strength reported by my laptop Network Manager is about 85%.

My next test is to set up an ISM band Sub-1GHz antenna to connect to my Sub 1GHz transceiver which will then talk with my receiver half to one km away.

This is my first field RF project, very exciting! 

I will use my building's public antenna.  As my target is civil ISM band under 20dbm radio power, I do not violate any radio rule.

You need to factor in that gain to know your radiated power.

So I have to use 50 Ohm coaxial cable, right?  They are quite expensive - 2 pounds per meter.

https://hk.element14.com/huber-suhner/enviroflex-b58/coaxial-cable-rg58-50-ohm-per/dp/2817372

Cable impedance is quite irrelevant since you have no info on the antenna impedance at the frequency you want to use. The antenna matcher will take care of your cable run to alleviate the SWR.

Ok, thanks for your opinions.

I am going to shop an antenna impedance tuner or solder one for my self.

I've found this post very informative.

www.ad5gg.com/.../

"I will use my building's public antenna."

"I have never seen our public antenna.  We cannot access our roof."

I don't think you understand; if you still mean the TV antenna, it's not a public "anyone can transmit" kind of thing. There's some electronics in-between the antenna and the end that you have access to. That electronics is there for amplification in one direction, for a particular frequency range. The electronics also renders it no good for transmit. And, as several people have mentioned, a TV antenna and its typical cable would be no good for LTE or any ISM band. 

In the city where signals are strong a simple passive multiway splitter can be used.

Bad news.  Public antenna is for receiving only.

Then, I have to set up my own antenna for testing. 

An antenna doesn't care whether its transmitting or receiving. They're just a cleverly spaced strip of conductive material. Essentially, antennas are designed to operate at specific frequencies. This means the length of the antenna has to match the wavelength. So have you confirmed whether the TV antenna is suitably matched to your desired transmit/receive frequency. Otherwise you will struggle to achieve the desired transmit distances, no matter the power.

Nowadays, TV is UHF range (around 400Mz) with digital transmission.

My target wireless connection target is an ISM band at 920MHz.

That is why I want to use my building`s public antenna. 

Unfortunately the antenna lengths required for the two frequencies aren't compatible - a 2cm difference is big for RF. According to omnicalculator.com dipole calculator (which gives a reasonable approximation and thus useful for comparison), the wavelengths don't match very well. As such it's unlikely to work at all, or at best you'll only achieve quite short TX distances.

Unfortunately the antenna lengths required for the two frequencies aren't compatible - a 2cm difference is big for RF. According to omnicalculator.com dipole calculator (which gives a reasonable approximation and thus useful for comparison), the wavelengths don't match very well. As such it's unlikely to work at all, or at best you'll only achieve quite short TX distances.