Musyoka's absence, Odinga told the thousands gathered at Uhuru Park to witness the occasion, was explainable and assured them that the former Vice President would be taking his oath as Deputy President, at a later date.
The East African country's Supreme Court annulled August's presidential election result saying the electoral commission committed "irregularities and illegalities" during the vote, harming the integrity of the election.
It would be recalled that Odinga had declared that he will not recognize a win by President Uhuru Kenyatta in the controversial repeat poll that was held previous year in October 26, which Odinga boycotted.
The Kenya Editor's Guild said in a statement Monday that President Uhuru Kenyatta had threatened to switch off TV stations that air live broadcast of the opposition's event.
Attorney General Githu Muigai warned last month that any "swearing-in" would be treasonous, raising the possibility of Odinga's arrest, a move that would heighten tensions.
"I'm happy this has happened but we are waiting for the next step forward but we don't know what it is", said nurse Grace Waithera.
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"Raila Odinga is quite popular in the poor areas and particular groupings so it provides a potential threat to internal stability, peace and security".
"We're trying to reach out informally to government officials, but we are not getting any answers", Waruru said.
As for plans following his unsanctioned inauguration ceremony, Odinga refused to reveal more, telling local news media to "hold your horses".
Though the police had said they would prohibit any illegal assemblies on Tuesday, there were no uniformed police in the park and no anti-riot officers or vehicles.
"If we do that without panicking in any way, we shall have peace because it doesn't mean someone will have an opportunity to take over the government by force", said Ole Sapit, according to Citizen TV.
Three main private television channels in the country had their live feeds cut or blocked, while the state-run Kenya Broadcasting Corporation (KBC) was allowed to continue.