Is it possible to measure a priori and a posteriori the effectiveness and profitability of an investment in training? How?

Yes, it's possible to measure the return on investment in training. Kirk Patrick put a model for this that has 4 different levels and which is widely used in organizations that has established training departments:

1. Level 1 - Feedback from participants

2. Level 2- Testing knowledge

3. Level 3 - Measuring business results

4. Level 4 - Measuring the monetary value of the return on investment. 

By measuring the difference between the productivity of the trainee before and after training

You can Increase the Quality of your training by using Evaluation forms for the trainee before and after the training, also after1 week or after1 month you can ask him if he applied what he learned in the workplace, to see the range of benefits and investments in these employees.

Thanks for invitation..... I prefer to wait for answers by professionals

Training is measurable but has 2 facets to see whether it is effective or not. If training is not measurable then am sure most of the organizations wouldn't have dedicated training functions to handle organizational training at different level.

1. Intangible results are those where your overall hygiene and organizations agendas are aligned by having modules which are appropriate for each level/function. These mostly for the alignment of organization across board which are driven as organizational culture requirement point of view.

2. Tangible are the 4 step process where you get feedback post training and you test the people of skills that you imparted to develop their skills, check business impact you had post developing their skills in respective areas of development and lastly the individual as an asset how much value added against the investment done.

It is very much measurable is what my sense.