It is impossible to prove the existence of God through scientific measurement. This includes anything which can be seen, heard, felt, tasted or smelled. At best, when endeavoring to convince our fellow man of His existence, we could find out what it would take for him or her to believe. This was the case with 'doubting Thomas':
...Except I shall see in his hands the print of the nails, and put my finger into the print of the nails, and thrust my hand into his side, I will not believe. John 20:25
In this situation, Jesus was able to grant his request. If one claims a miraculous happening is from God, and another sees it for him or herself, this may be enough for the person to believe, but the happening will still not be the kind of proof that can be used in a courtroom.
After Thomas was convinced, Jesus said: "blessed are they that have not seen, and yet have believed." Seeing a miracle is not a necessary prerequisite for a person to believe. If we want to see the people we care about in heaven, the best we can do for them is to make 'proclaiming of the truth' part of everything we do:
Holding forth the word of life; that I may rejoice in the day of Christ, that I have not run in vain, neither laboured in vain. Philippians 2:16
Once adopting this way of life, a believer will find that it is possible to prove God exists, but only to him or herself, and to those willing to believe.
And they went forth, and preached every where, the Lord working with them, and confirming the word with signs following. Amen. Mark 16:20