Can we calculate the average temperature of the heliosphere? It is my understanding that one can measure the temperature of the universe by measuring cosmic microwave background radiation. Could we also use this method for calculating the average temperature of the heliosphere?

The average temperature of the universe has supposedly been measured in at 2.735 degrees above absolute zero. Presumably this measurement takes into account the massive amounts of 'empty' space which I imagine has a huge effect on the number. 

My question: Can we measure the average temperature of the heliosphere? If so, then what is it? 

This might be quite hard to define as how do you determine when our solar system ends? So for the purpose of this question the following assumptions should be made:


The distance from the Sun to the edge of the heliosphere is approximately 100AU (Roughly the size of the heliosphere as measured by Voyager 1).
Assume the heliosphere is a perfect sphere.
Assume we are measuring the black body equivalent temperature (as this is what was measured to calculate the temperature of the universe).


If we don't know how to measure this average temperature (which is likely if we can't differentiate between microwaves from in and outside the heliosphere) then how could we theoretically measure this value assuming a perfect environment?
The heliosphere is mainly defined by the region dominated by solar wind against the interstellar medium.

"The solar wind is divided into two components, respectively termed the slow solar wind and the fast solar wind. The slow solar wind has a velocity of about 400 km/s, a temperature of $1.4–1.6×10^6 K$ and a composition that is a close match to the corona. By contrast, the fast solar wind has a typical velocity of 750 km/s, a temperature of $8×10^5 K$ and it nearly matches the composition of the Sun's photosphere."

Hence with 1 million Kelvin we get a good estimate of the average temperature of the heliosphere.

The temperature is the freeze-in temperature derived from carbon ion charge states measured by SOHO. Hence it's not the black-body temperature, but a reasonable definition of temperature for solar wind.

"Coronal temperatures are inferred from density ratios of adjacent charge states. The freeze-in temperature derived from a given density ratio is the electron temperature that reproduces this ratio in a static situation. For the analyzed iron charge states we use the ionization and recombination rates of Arnaud and Raymond [1992]". More details about SOHO's temperature measurement here.

If the heliosphere would be in the thermodynamic equilibrium, according to Planck's law or Wien's law, you would get the peak at about 3nm, in the soft x-ray range. But the equilibrium assumption probably doesn't hold.

For the hot interstellar medium extreme ultraviolet radiation measurements have been tried (CHIPSat), some successful (EUVE).
Hence, in an ideal environment we would measure the radiation intensity across the electromagnetic spectrum, look for the peak intensity, apply Wien's law, and retrieve the  effective temperature, which would be identical to the black body temperature in an ideal world.

The plasma of the heliosphere is very rare, therefore almost transparent. That's why we don't feel that heat of about 1 milion degrees. Microwaves play a minor role in that temparature range.

A temperature profile of Sun's corona and heliosphere is shown e.g. on pages 42 and 47 of this survey.